2007-12-66

NEWSLETTER
OF THE EUROPEAN MATHEMATICAL SOCIETY
3 9 6
2 8
7 4
7
5
2
3
Feature
Sudoku puzzles
History
Euler in Berlin
Interview
Beno Eckmann
Societies
Cyprus Mathematical Society
p. 13
p. 21
p. 31
p. 41
December 2007
Issue 66
ISSN 1027-488X
S
M
E
E
M
S
European
Mathematical
Society

Oxford University Press is pleased to announce that all EMS members can benefit
from a 20% discount on a large range of our Mathematics books.
For more information please visit: http://www.oup.co.uk/sale/science/ems
Mathematics Books from Oxford
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Foliations and the Geometry of
3-Manifolds
Danny Calegari
This unique reference, aimed at research
topologists, gives an exposition of the 'pseudo-
Anosov' theory of foliations of 3-manifolds.
OXFORD MATHEMATICAL MONOGRAPHS
May 2007 | 378 pp | Hardback
978-0-19-857008-0 | £60.00
Advanced Distance Sampling
Estimating abundance of biological
populations
Edited by S.T. Buckland, D.R Anderson,
K.P. Burnham, J.L. Laake,
D.L. Borchers, and L.
Thomas
Reconstructing Evolution
New Mathematical and
Computational Advances
Edited by Olivier Gascuel and Mike Steel
Evolution is a complex process, acting at multiple
scales, from DNA sequences and proteins to
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June 2007 | 348 pp | Hardback | 978-0-19-920822-7 | £39.50
Flips for 3-folds and 4-folds
Edited by Alessio Corti
Aimed at graduates and researchers in algebraic
geometry, this collection of edited chapters provides
a complete and essentially self-contained account of
the construction of 3-fold and 4-fold klt flips.
OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS NO. 35
June 2007 | 200 pp | Hardback
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Mathematics of Evolution and
Phylogeny
Olivier Gascuel
This book of contributed chapters is authored by
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new in paperback
October 2007 | 442 pp | Paperback | 978-0-19-923134-8 | £24.95
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This advanced text focuses on the uses of distance
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Nov 2007 | 434 pp | Paperback | 978-0-19-922587-3 | £24.95
August 2004 | 434 pp | Hardback| 978-0-19-850783-3 | £59.00
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Atmospheric Turbulence
A Molecular Dynamics Perspective
Adrian Tuck
This book focuses on the direct link between
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January 2008 | 168 pp | Hardback
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Partial Differential Equations in
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Invitation to Discrete Mathematics 2/e
Jiøí Matoušek and Jaroslav Nešetøil
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Godfrey H. Hardy and Edward M. Wright
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Editorial Team
Editor-in-Chief
Martin Raussen
Department of Mathematical
Sciences
Aalborg University
Fredrik Bajers Vej 7G
DK-9220 Aalborg Øst,
Denmark
e-mail: raussen@math.aau.dk
Associate Editors
Vasile Berinde
Department of Mathematics
and Computer Science
Universitatea de Nord
Baia Mare
Facultatea de Stiinte
Str. Victoriei, nr. 76
430072, Baia Mare, Romania
e-mail: vberinde@ubm.ro
Krzysztof Ciesielski
(Societies)
Mathematics Institute
Jagellonian University
Reymonta 4
PL-30-059, Kraków, Poland
e-mail: Krzysztof.Ciesielski@im.uj.edu.pl
Robin Wilson
Department of Mathematics
The Open University
Milton Keynes, MK7 6AA, UK
e-mail: r.j.wilson@open.ac.uk
Copy Editor
Chris Nunn
4 Rosehip Way
Lychpit
Basingstoke RG24 8SW, UK
e-mail: nunn2quick@qmail.com
Editors
Giuseppe Anichini
Dipartimento di Matematica
Applicata “G. Sansone”
Via S. Marta 3
I-50139 Firenze, Italy
e-mail: giuseppe.anichini@unifi .it
Chris Budd
(Applied Math./Applications
of Math.)
Department of Mathematical
Sciences, University of Bath
Bath BA2 7AY, UK
e-mail: cjb@maths.bath.ac.uk
Mariolina Bartolini Bussi
(Math. Education)
Dip. Matematica - Universitá
Via G. Campi 213/b
I-41100 Modena, Italy
e-mail: bartolini@unimo.it
Ana Bela Cruzeiro
Departamento de Matemática
Instituto Superior Técnico
Av. Rovisco Pais
1049-001 Lisboa, Portugal
e-mail: abcruz@math.ist.utl.pt
Vicente Muñoz
(Book Reviews)
IMAFF – CSIC
C/Serrano, 113bis
E-28006, Madrid, Spain
vicente.munoz @imaff.cfmac.csic.es
Ivan Netuka
(Recent Books)
Mathematical Institute
Charles University
Sokolovská 83
186 75 Praha 8
Czech Republic
e-mail: netuka@karlin.mff.cuni.cz
Mădălina Păcurar
(Conferences)
Department of Statistics,
Forecast and Mathematics
Babeș-Bolyai University
T. Mihaili St. 58-60
400591 Cluj-Napoca, Romania
mpacurar@econ.ubbcluj.ro;
madalina_pacurar@yahoo.com
Frédéric Paugam
Institut de Mathématiques
de Jussieu
175, rue de Chevaleret
F-75013 Paris, France
e-mail: frederic.paugam@math.jussieu.fr
Ulf Persson
Matematiska Vetenskaper
Chalmers tekniska högskola
S-412 96 Göteborg, Sweden
e-mail: ulfp@math.chalmers.se
Walter Purkert
(History of Mathematics)
Hausdorff-Edition
Mathematisches Institut
Universität Bonn
Beringstrasse 1
D-53115 Bonn, Germany
e-mail: edition@math.uni-bonn.de
Themistocles M. Rassias
(Problem Corner)
Department of Mathematics
National Technical University
of Athens
Zografou Campus
GR-15780 Athens, Greece
e-mail: trassias@math.ntua.gr.
Vladimír Souček
(Recent Books)
Mathematical Institute
Charles University
Sokolovská 83
186 75 Praha 8
Czech Republic
e-mail: soucek@karlin.mff.cuni.cz
Contents
European
Mathematical
Society
Newsletter No. 66, December 2007
EMS Calendar ............................................................................................................................. 2
Editorial ................................................................................................................................................. 3
EMS-Council meeting 2008 .................................................................................... 7
EMS Committee on Developing Countries ........................................ 9
Dubna Summer School – M. Eickenberg .............................................. 11
Sudoku puzzles – A. Brouwer ............................................................................. 13
Paulette Libermann (1919–2007) – M. Audin ................................... 19
Euler in Berlin – J. Brüning ..................................................................................... 21
Interview with Beno Eckmann – M. Raussen & A. Valette ...... 31
A Survey of ICMI Activities – M. Bartolini Bussi .......................................... 37
Cyprus Mathematical Society – G. Makrides ................................. 41
Personal Column .................................................................................................................... 44
Book Review: Music and Mathematics – X. Gràcia .............. 45
Forthcoming Conferences ........................................................................................ 46
Recent Books .............................................................................................................................. 53
The views expressed in this Newsletter are those of the
authors and do not necessarily represent those of the
EMS or the Editorial Team.
ISSN 1027-488X
© 2007 European Mathematical Society
Published by the
EMS Publishing House
ETH-Zentrum FLI C4
CH-8092 Zürich, Switzerland.
homepage: www.ems-ph.org
For advertisements contact: newsletter@ems-ph.org
EMS Newsletter December 2007 1

EMS News
EMS Executive Committee
EMS Calendar
President
Prof. Ari Laptev
(2007–10)
Department of Mathematics
South Kensington Campus,
Imperial College London
SW7 2AZ London, UK
e-mail:a.laptev@imperial.ac.uk
and
Department of Mathematics
Royal Institute of Technology
SE-100 44 Stockholm, Sweden
e-mail: laptev@math.kth.se
Vice-Presidents
Prof. Pavel Exner
(2005–08)
Department of Theoretical
Physics, NPI
Academy of Sciences
25068 Rez – Prague
Czech Republic
e-mail: exner@ujf.cas.cz
Prof. Helge Holden
(2007–10)
Department of Mathematical
Sciences
Norwegian University of
Science and Technology
Alfred Getz vei 1
NO-7491 Trondheim, Norway
e-mail: h.holden@math.ntnu.no
Secretary
Dr. Stephen Huggett
(2007–10)
School of Mathematics
and Statistics
University of Plymouth
Plymouth PL4 8AA, UK
e-mail: s.huggett@plymouth.ac.uk
Treasurer
Prof. Jouko Väänänen
(2007–10)
Department of Mathematics
and Statistics
Gustaf Hällströmin katu 2b
FIN-00014 University of Helsinki
Finland
e-mail: jouko.vaananen@helsinki.fi
and
Institute for Logic, Language
and Computation
University of Amsterdam
Plantage Muidergracht 24
1018 TV Amsterdam
The Netherlands
e-mail: vaananen@science.uva.nl
Ordinary Members
Prof. Victor Buchstaber
(2005–08)
Steklov Mathematical Institute
Russian Academy of Sciences
Gubkina St. 8
Moscow 119991, Russia
e-mail: buchstab@mi.ras.ru and
Victor.Buchstaber@manchester.ac.uk
Prof. Olga Gil-Medrano
(2005–08)
Department de Geometria i
Topologia
Fac. Matematiques
Universitat de Valencia
Avda. Vte. Andres Estelles, 1
E-46100 Burjassot, Valencia
Spain
e-mail: Olga.Gil@uv.es
Prof. Mireille Martin-
Deschamps
(2007–10)
Département de
mathématiques
45, avenue des Etats-Unis
Bâtiment Fermat
F-78030 Versailles Cedex,
France
e-mail: mmd@math.uvsq.fr
Prof. Carlo Sbordone
(2005–08)
Dipartmento de Matematica
“R. Caccioppoli”
Università di Napoli
“Federico II”
Via Cintia
80126 Napoli, Italy
e-mail: carlo.sbordone@fastwebnet.it
Prof. Klaus Schmidt
(2005–08)
Mathematics Institute
University of Vienna
Nordbergstrasse 15
A-1090 Vienna, Austria
e-mail: klaus.schmidt@univie.ac.at
EMS Secretariat
Ms. Riitta Ulmanen
Department of Mathematics
and Statistics
P.O. Box 68
(Gustaf Hällströmin katu 2b)
FI-00014 University of Helsinki
Finland
Tel: (+358)-9-1915-1426
Fax: (+358)-9-1915-1400
Telex: 124690
e-mail: ems-offi ce@helsinki.fi
Web site: http://www.emis.de
2008
1 January
Deadline for the proposal of satellite conferences to 5ECM
top@math.rug.nl; www.5ecm.nl/satellites.html
1 February
Deadline for submission of nominations of candidates for the
Felix Klein prize
ems-offi ce@cc.helsinki.fi ; www.emis.de/tmp2.pdf
1 February
Deadline for submission of material for the March issue of the
EMS Newsletter
Martin Raussen: raussen@math.aau.dk
29 February–2 March
Joint Mathematical Weekend EMS-Danish Mathematical
Society, Copenhagen (Denmark)
www.math.ku.dk/english/research/conferences/emsweekend/
3 March
EMS Executive Committee Meeting at the invitation of the
Danish Mathematical Society, Copenhagen (Denmark)
Stephen Huggett: s.huggett@plymouth.ac.uk
30 June–4 July
The European Consortium For Mathematics In Industry (ECMI),
University College London (UK)
www.ecmi2008.org/
11 July
EMS Executive Committee Meeting, Utrecht (The Netherlands)
Stephen Huggett: s.huggett@plymouth.ac.uk
12–13 July
EMS Council Meeting, Utrecht (The Netherlands)
Stephen Huggett: s.huggett@plymouth.ac.uk
Riitta Ulmanen: ems-offi ce@cc.helsinki.fi
13 July
Joint EWM/EMS Workshop, Amsterdam (The Netherlands)
http://womenandmath.wordpress.com/joint-ewmemsworskhop-amsterdam-july-13th-2008/
14–18 July
5th European Mathematical Congress, Amsterdam
(The Netherlands)
www.5ecm.nl
3–9 August
Junior Mathematical Congress 8, Jena (Germany)
www.jmc2008.org/
16–31 August
EMS-SMI Summer School at Cortona (Italy)
Mathematical and numerical methods for the cardiovascular
system
dipartimento@matapp.unimib.it
8–19 September
EMS Summer School at Montecatini (Italy)
Mathematical models in the manufacturing of glass, polymers
and textiles
web.math.unifi .it/users/cime//
2 EMS Newsletter December 2007

5ECM:
14–18 July 2008
in Amsterdam
Editorial
André Ran, Herman te Riele and Jan Wiegerinck
In a little more than half a year we will be enjoying
5ECM. It is going to be a fantastic event for European
mathematics. Over the period 14–18 July 2008 the RAI
Convention Center in Amsterdam will be the meeting
point of mathematicians from all over Europe.
The Scientific Committee, chaired by Lex Schrijver,
has put together a wonderful program. With ten plenary
speakers, over 30 invited presentations by outstanding
mathematicians on the newest developments in both
pure and applied mathematics, three science lectures and
over twenty mini-symposia, the programme is already of
a high level. In addition there will of course be the ten
young winners of the EMS prizes. The programme is an
excellent mix of various areas of mathematics, both pure
and applied. All in all, this is an event to look forward to.
Do come and join us in Amsterdam next year!
The website http://www.5ecm.nl is growing rapidly
with information on the lectures, the mini-symposia and
much more relevant information.
For EMS members the conference fee is a very modest
220 Euro when registering before 01 April 2008 (note
that this fee is lower than the registration fee for the
ICIAM meeting last year, which amounted to 240 Euro!).
The organisers acknowledge generous support from the
NWO (the Dutch Science Foundation), the KWG (the
Royal Mathematical Society of The Netherlands) and
the Foundation Compositio Mathematica. Several Dutch
companies have also supported the conference. At
present the organisers are still working extremely hard
to gather more support for the conference, in particular
for funds to support the participants.
The plenary lecturers
To convince you that it is worthwhile to come to Amsterdam
next summer for an excellent conference, we will introduce
the ten plenary lecturers below followed by the
three science lecturers. We are confident that with such an
extraordinary group of main lecturers you will not want
to miss the fifth European Congress of Mathematics.
Luigi Ambrosio (Scuola Normale Superiore di Pisa)
Born in Italy in 1963, Ambrosio obtained his PhD in 1985
in Pisa under the guidance of E. De Giorgio. He has previously
been a speaker at the ECM in 1996 and at the
ICM in 2002. He was a winner of the Fermat
Prize of the University of Toulouse in
2003 and several other prizes of the Unione
Matematica Italiana.
He is one of the leading experts in the
field of calculus of variations. He has discovered
some important results, including
the introduction of SBV spaces, the theory of currents,
mass transport, and rectifiable sets.
Christine Bernardi (CNRS and Université
Pierre et Marie Curie Paris)
Born in France in 1955, Christine Bernardi
obtained her PhD in 1979. She was awarded
the 1995 Prix Blaise Pascal of the GAMNI-
SMAI of the French Academy of Sciences.
She is a member of the top group studying
Numerical Analysis in Europe. She
and her group have made remarkable contributions to
several fields in numerical analysis, from spectral methods
to domain decomposition methods, to more recent
and interesting techniques such as the reduced-bases
method and some new applications of a posteriori analysis.
Jean Bourgain (IAS Princeton)
Born in Belgium in 1954, Bourgain obtained
his PhD in Brussels in 1977 in Banach
space theory. He was winner of the
Fields medal in 1994, the Ostrowski Prize
in 1991 and of many other awards and distinctions.
He was an invited speaker at the ICM in 1983,
1986 and 1994, and at the ECM in 1992.
He has published more than 300 papers in several
branches of mathematics, including analysis (Banach
spaces, harmonic analysis), analytic number theory, combinatorics,
ergodic theory and partial differential equations.
Jean-François Le Gall
(Université Paris-Sud, Orsay)
Born in France in 1959, Le Gall obtained
his PhD in 1982 on probability theory
under the supervision of Marc Yor. He
was a winner of the 1997 Loève Prize in
Probability, the 2005 Fermat Prize and the Grand Prix
EMS Newsletter December 2007 3

Editorial
Sophie Germain de l’Académie des Sciences de Paris in
the same year. He was an invited speaker at the ECM in
1992 and at the ICM in 1998. He was also the thesis advisor
of the 2006 Fields medallist Wendelin Werner.
His area of research is probability theory, including
Brownian motion, Lévy processes, superprocesses and
their connections with partial differential equations, the
Brownian snake, random trees, branching processes, and
coalescence. Particularly interesting is his recent work on
the scaling limits of random planar maps. This is in the
intersection of probability, combinatorics and statistical
physics.
François Loeser (ENS Paris)
Born in France in 1958, Loeser obtained
his PhD in 1983 under the supervision of
B. Teissier. He was a plenary speaker at
the 2005 AMS Algebraic Summer School
in Seattle. In 2007 he was awarded the Prix
Charles-Louis de Saulces de Freycinet of
the French Academy of Sciences.
His research areas are algebraic geometry, arithmetic
geometry and motivic integration.
László Lovász
(Eötvös Loránd University, Budapest)
Born in Hungary in 1948, Lovász received
his PhD in 1971 under the supervision
of Tibor Gallai. He was a collaborative
member of the Microsoft Research Redmond
lab. (WA, USA) until 2006. He has
received many awards and honours, among them the
Wolf Prize and the Knuth Prize, and he was a plenary
speaker at the ICM in 1990. Recently he was awarded
the Bolyai Prize.
His research focuses on discrete mathematics, optimization,
complexity and algorithms. His name is connected
to several problems and results in mathematics: in
graph theory, there are the Erdős-Faber-Lovász conjecture
about the colouring of graphs and the Lovász conjecture
concerning Hamiltonian paths in certain classes
of graphs; and in the theory of algorithms there is the
Lenstra-Lenstra-Lovász lattice basis reduction (LLL) algorithm.
Matilde Marcolli
(Max Planck Institut Bonn)
Born in Italy in 1969, Marcolli obtained
her PhD in 1997 in Chicago under the supervision
of M. Rotherberg. She was a recipient
of the Sofia Kowalewskaya Award
and the Heinz-Maier-Leibnitz-Preis of
the DFG. She is editor of the European Mathematical
Society Publishing House’s “Journal of Noncommutative
Geometry”.
Her research is concerned with noncommutative geometry,
applications to physics, number theory and differential
geometry. Marcolli is one of the most active and
original present-day mathematicians of Europe.
Felix Otto (Universität Bonn)
Born in Germany in 1966, Otto obtained
his PhD in Bonn in 1993 under guidance
of S. Luckhaus. After that he spent several
years in the USA, at Courant Institute,
Carnegie Mellon University and UC Santa
Barbara, before returning to Bonn in
1999. He has given invited lectures at several important
conferences (SIAM, GAMM, ICM sectional speaker in
2002, ICIAM 2007) and was Stieltjes Visiting Professor in
the Netherlands in 2004. He was the recipient of the 1997
A. P. Sloan Research Fellowship, the 2001 Max-Planck
Forschungspreis, the 2006 Gottfried-Wilhelm-Leibniz-
Preis and the 2007 Collatz Prize.
His research topics are centred around the analysis of
pattern formation in models from physics, partial differential
equations and multi-scale methods. The methods
of analysis used in his group are wide ranging, including
numerical simulation, asymptotic analysis and rigorous
analysis. Micromagnetism, complex rheology and coarsening
of spatial structure in time are just a few topics of
recent research in his group.
Nicolai Reshetikhin
(University of California, Berkeley)
Born in Russia in 1958, Reshetikhin obtained
his PhD in 1984 at LOMI in his
home town Leningrad. He was a speaker
at the ICM in 1990. Presently he is at UC
Berkeley and at Aarhus, where he holds
the Niels Bohr professorship for the years 2006–2010.
Reshetikhin is working in the overlapping areas of geometry,
topology and mathematical physics. He has had
great influence in a number of mathematical disciplines,
including representation theory, knot theory and moduli
spaces of flat connections. His work with V. Turaev on topological
quantum field theories in dimension three has had
a huge influence on lots of young researchers – it opened
up the subject of what could be called Quantum Topology.
Richard Taylor
(Harvard University, Cambridge)
Born in England in 1962, Taylor obtained
his PhD in 1988 in Princeton under the
supervision of Andrew Wiles. He received
the Fermat Prize 2001, the Ostrowski
Prize 2001, the Cole Prize 2002 of the
American Mathematical Society and the
Shaw Prize 2007 in Mathematical Sciences for his work
on the Langlands program. He has been a speaker at the
ICM meetings in 1994 and 2002.
Taylor is perhaps best known for his contribution to the
solution of Fermat’s last problem, work he did with Andrew
Wiles several years after receiving his PhD. His research
interests are algebraic number theory, modular forms and
Galois representations. He is considered to be the prime
expert on the recent progress on the interaction of automorphic
forms and arithmetic geometry, in particular the
Sato-Tate conjecture and Serre’s modular conjecture.
4 EMS Newsletter December 2007

Editorial
The three Science lecturers
Juan Ignacio Cirac (Garching, Germany)
on Quantum Information Theory
Juan Ignacio Cirac is an internationally
renowned leader in quantum information
science with an extraordinary broad
scope in the field, ranging from theoretical
aspects to physical realizations.
He graduated from the Universidad Complutense de
Madrid in 1988 and moved to the United States in 1991
to work as a postdoctoral scientist in the Joint Institute
for Laboratory Astrophysics in the University of Colorado
at Boulder. Later he became a professor at the Institut
für Theoretische Physik in Innsbruck, Austria. He
has been the Director of the Theoretical Division of the
Max Planck Institute for Quantum Optics in Garching,
Germany, since 2001.
His research is on the quantum theory of information.
He has developed a computational system based on
quantum mechanics that is hoped to lead to the design of
much faster algorithms in the future. He has contributed
to applications that demonstrate the viability of his theories,
performing calculations that are impossible with
current systems for processing and transmitting information.
According to his theories, quantum computing will
revolutionise the information society and lead to much
more efficient and secure communication of information.
Apart from his interest in quantum theory, he has
investigated degenerated quantum gases and quantum
optics. He is one of the most cited authors in his field of
research.
Tim Palmer (Reading, UK) on
Climate Change
Tim Palmer has been working at the European
Centre for Medium-Range Weather
Forecasts in Reading since the mid 1980s,
where he now heads up the Predictability
and Seasonal Forecast Division. He is a
Fellow of the Royal Society.
He has contributed significantly to the application
of nonlinear mathematical methods to understanding
non-trivial aspects of global warming. His team develops
complicated models of the Earth system that are based
on the laws of physics and comprise tens of millions of
equations. Forecasts from this model are fed to most of
the National Meteorological Services around Europe.
Palmer has been applying ensemble forecast techniques
to longer timescales, leading a team that develops predictions
through coupled ocean-atmosphere models. These
models can, for example, give risk predictions of the
failure of the monsoon rains months in advance. On an
even longer timescale, he has been involved with questions
concerning global warming. In addition, Palmer is
engaged in interdisciplinary research on forecasting rare
events like malaria epidemics, river flooding and crop
failure (all weather related). The ultimate goal of such
research would be to target, for instance, resources to
help prevent a malaria epidemic in the regions that are
most at risk.
Jonathan Sherratt (Edinburgh, UK) on
Mathematical Biology
Jonathan Sherratt has been a professor of
mathematics at Heriot-Watt University
since January 1998. He was elected as a
Fellow of the Royal Society of Edinburgh
in 2001. He has been very prominent since
his first work on wound healing. In 2006 he was awarded
the Whitehead Prize of the London Mathematical Society
for his contributions to mathematical biology and, in
particular, the development and analysis of new mathematical
models for complex biological processes.
He has worked on a broad range of topics. Notable
among these are wound healing, pattern formation, tumour
growth and spatiotemporal chaos. His work is truly
multidisciplinary and often collaborative, and is characterized
by originality and novelty. He has published
widely in mathematical biology journals and the biological
literature. His papers are motivated by biological and
medical problems, applying modelling and analysis to
understand these, and then providing a significant link
back to biology.
André Ran, Herman te Riele and Jan Wiegerinck are the
members of the local organising committee of 5ECM.
Groups, Geometry, and Dynamics
ISSN: 1661-7207
2008. Vol. 2, 4 issues
Approx. 600 pages.17.0 cm x 24.0 cm
Price of subscription, including electronic edition: 212 Euro plus shipping (add 20 Euro for normal delivery). Other subscriptions on request.
Aims and Scope: Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as
articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory
with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions.
Editor-in-Chief: Rostislav Grigorchuk (USA and Russia)
Managing Editors: Tatiana Smirnova-Nagnibeda (Switzerland), Zoran S˘ unić (USA)
Editors: Gilbert Baumslag (USA), Mladen Bestvina (USA), Martin R. Bridson (UK), Tullio Ceccherini-Silberstein (Italy), Anna Erschler (France), Damien Gaboriau (France), Étienne Ghys
(France), Fritz Grunewald (Germany), Tadeusz Januszkiewicz (USA and Poland), Michael Kapovich (USA), Marc Lackenby (UK), Wolfgang Lück (Germany), Nicolas Monod (Switzerland),
Volodymyr Nekrashevych (USA and Ukraine), Mark V. Sapir (USA), Yehuda Shalom (Israel), Dimitri Shlyakhtenko (USA), Efim Zelmanov (USA).
European Mathematical Society Publishing House
Seminar for Applied Mathematics, ETH-Zentrum FLI C4
Fliederstrasse 23
CH-8092 Zürich, Switzerland
subscriptions@ems-ph.org
www.ems-ph.org
EMS Newsletter December 2007 5

EMS News
EMS Council Meeting, Utrecht 2008
Pelczar, Andrzej, 2000–2003–2007
Sodin, Mikhail, 2004–2007–2011
Soifer, Gregory, 2004–2007–2011
Tronel, Gerard, 2000–2003–2007
Wilson, Robin, 2004–2007–2011
Xambo-Descamps, Sebastian, 2002–2005–2009
Academiegebouw Utrecht.
The EMS Council meets every second year. The next
meeting will be held in Utrecht, the Netherlands, July 12
and 13, 2008 in the Academiegebouw of Utrecht University,
before the 5ECM conference in Amsterdam, 14–18
July, 2008. The Council meeting starts at 13.00 on July 12
and ends at noon on July 13.
Delegates to the Council will be elected by the following
categories of members.
(a) Full Members
Full Members are national mathematical societies, which
elect 1, 2, or 3 delegates according to their membership
class. Each society is responsible for the election of its
delegates. Each society should notify the Secretariat of
the EMS in Helsinki of the names and addresses of its
delegate(s) no later than 28 February 2008. As of 1 July
2007, there were 57 such societies, which could designate
a maximum of 86 delegates.
(b) Institutional Members
Delegates shall be elected for a period of four years. A
delegate may be re-elected provided that consecutive
service in the same capacity does not exceed eight years.
The institutional members, as a group, nominate candidates
and elect delegates.
(c) Individual Members
On 30 June 2007, there were 2306 individual members
and, according to our statutes, these members will be represented
by (up to) 24 delegates.
The present delegates of individual members are:
Anichinii, Giuseppe, 2006–2009
Berinde, Vasile, 2000–2003–2007
Anzellotti, Gabriele, 2004–2007–2011
Conte, Alberto, 2000–2003–2007
Coti Zelati, Vittorio, 2004–2007–2011
Guillopé, Laurent, 2000–2003–2007
Higgs, Russell, 2004–2007–2011
Kingman, John, 2006–2009
Margolis, Stuart W, 2004–2007–2011
The mandates of Berinde, Conte, Guillopé, Pelczar, and
Tronel end on 31 December 2007. They cannot be reelected
because they have served in this capacity for eight
years. A nomination form for proposals of new delegates
of individual members is attached below. Please note the
deadline 28 February 2008 for nominations to arrive in
Helsinki. Elections of individual delegates will be organised
by the EMS secretariat by postal ballot among individual
members unless the number of nominations does
not exceed the number of vacancies.
Agenda
The Executive Committee is responsible for preparing
the matters to be discussed at Council meetings. Items
for the agenda of this meeting of the Council should be
sent as soon as possible, and no later than 28 February
2008, to the EMS Secretariat in Helsinki.
Executive Committee
The Council is responsible for electing the President,
Vice-Presidents, Secretary, Treasurer and other members
of the Executive Committee. The present membership of
the Executive Committee, together with their individual
terms of office, is as follows.
President: Professor A. Laptev (2007–2010)
Vice-Presidents: Professor P. Exner (2005–2008)
Professor H. Holden (2007–2010)
Secretary: Dr S. Huggett (2007–2010)
Treasurer: Professor J. Väänänen (2007–2010)
Members: Professor V. Buchstaber (2001–2004–2008)
Professor O. Gil-Medrano (2005–2008)
Professor M. Martin-Deschamps (2007–2010)
Professor C. Sbordone (2005–2008)
Professor K. Schmidt (2005–2008)
Members of the Executive Committee are elected for a
period of four years. The President can only serve one
term. Committee members may be re-elected, provided
that consecutive service shall not exceed eight years. One
position of vice-president and four positions of membersat-large
of the Executive Committee will be vacant as of
31 December 2008; elections will arise during Council.
V. Buchstaber has served on the Executive Committee
for eight years and cannot be re-elected. Nominations of
candidates are invited; more information below.
The Council may, at its meeting, add to the nominations
received and set up a Nominations Committee,
disjoint from the Executive Committee, to consider all
EMS Newsletter December 2007 7

EMS News
candidates. After hearing the report by the Chair of the
Nominations Committee (if one has been set up), the
Council will proceed to the elections to the Executive
Committee posts.
All these arrangements are as required in the Statutes
and By-Laws. All information and material concerning
the Council will be made available at www.math.ntnu.
no/ems/council08.
Secretary: Stephen Huggett (s.huggett@plymouth.ac.uk)
Secretariat: Riitta Ulmanen (ems-office@helsinki.fi)
Elections to the Executive Committee:
Nominations
At the end of 2008 there will be vacancies for the position
of one Vice-President, as well as of four ordinary
members of the Executive Committee.
Nominations for these positions are invited. Any
member of the Society may be nominated. It would be
convenient if nominations for office in the Executive
Committee, duly signed and seconded, could reach the
Secretariat by 28 February, 2008. It is strongly recommended
that a statement of intention or policy is enclosed
with each nomination. If the nomination comes from the
floor during the Council meeting there must be a clear
declaration of the willingness of the person to serve. It is
recommended that statements of policy of the candidates
nominated from the floor should be available.
Nomination Form for Council Delegate
Name: ………………......……………………………………………………….
Title: …………………………………………..…………………………………
Address: ……………………………………….………………………………
………………………………………………………………………………………
Proposed by: ………………………………...………………………………
Seconded: ……………………………..………………………………………
I certify that I am an individual member of the
EMS and that I am willing to stand for election as a
delegate of individual members at Council.
Signature of Candidate: .....……………………………………………
Date: …………………………………………………….………………………
Completed forms should be sent to:
Riitta Ulmanen
EMS Secretariat
Department of Mathematics and Statistics
P.O.Box 68
FI-00014 University of Helsinki
Finland
to arrive by 28 February, 2008. A photocopy of this
form is acceptable.
8 EMS Newsletter December 2007

EMS-CDC – Alive and kicking
EMS News
Some of the readers of the EMS Newsletter might have
got the impression during the last couple of years that
the EMS Committee for Developing Countries (EMS-
CDC for short) has faded away. Don’t worry, EMS-CDC
is alive and spreading its wings in line with its policy
statement: see http://www.emis.de/committees.html#dc
There was a short period in 2004/05 when our most
important activity so far – our book donation programme
– came to a temporary halt because of a shortage
of funds needed to cover the shipping expenses of
scientific literature from the donors to the recipients.
In fact, ICTP which had generously supported this programme
could no longer do so. EMS-EC bailed us out,
and then the current EMS-CDC chair, in her capacity as
one of the editors of the Encyclopaedia of Mathematical
Physics (Elsevier 2006), called upon the authors to
donate their honorariums to EMS-CDC, which many of
them did. Here we want to say once more ‘Thank you’
to these authors. As a consequence, our Book Donation
Programme is running smoothly ever since. This also because
we have joined forces with AMS in this direction.
Moreover, we have enlarged the scope of the book donation
programme by producing CDs of lecture notes in
English, for the exclusive use by colleagues in developing
countries. It is intended to make CDs of lecture notes in
French and Spanish to make it easier for students and
colleagues of francophone developing countries and of
Latin American countries, respectively.
We had several rezquests for financial support of conferences
(notably ICM 2006 in Madrid, and a conference
in Mongolia), and also workshops, which we were able to
fulfil. However, we can do this only in exceptional cases
precisely because of our relative shortage of finances. We
are also supporting a Cambodian student to do a master’s
degree in Vietnam. We note that the situation in Cambodia
is extremely poor for mathematics: there is only one
Ph.D. in mathematics in the whole country. The IMU and
CIMPA are organizing series of lectures there and we
intend to join forces with them.
Moreover, we are currently interacting with the University
of KwaZulu Natal (Durban/Westville campus) in
developing a proof oriented M.Sc. programme in mathematics.
This programme is ultimately intended to become
a regional programme for students of southern if
not sub-Saharan Africa. This programme is based on the
experience gathered in developing countries that programmes
in mathematical modelling are very often focusing
too much on teaching techniques of how to solve
this or that problem and (necessarily) neglecting at the
same time to prove basic mathematical results. On the
other hand, universities in many developing countries
are facing a permanent shortage of academic staff. Thus,
proof oriented M.Sc. programmes facilitate the training
of academic staff who have a solid training in proving
mathematical results in various mathematical areas.
Several of our EMS-CDC members are heads of development
organizations and are directly involved in
various programmes in developing countries. Very often
such programmes are independent of the existence of
EMS-CDC. This makes it even easier for us to collaborate
and to assist such programmes – partly financially,
partly by manpower. Thus we have enlarged our scope
of activities also in this direction, by supervising students
both at M.Sc. and Ph.D. level.
As before, we are closely cooperating with various
mathematical organizations, some of which (like CIM-
PA, but also various national mathematical societies) are
represented directly or indirectly by members of EMS-
CDC. This results in useful information flowing in both
directions, but also support of EMS-activities such as the
book donation programme.
There seems to be a growing awareness among mathematicians
in developed countries that their help is needed
in developing countries. Taking this growing readiness
to assist maths departments in developing countries, it
was proposed at one of our annual meetings that universities
in developed countries could “adopt” a university
in a developing country and pay, for example, for the
subscription of Zentralblatt or at least its online version,
or for MathSciNet (both Zentralblatt and Math Reviews
have special rates for developing countries – in many
cases the rate is only 10% of the list price). This would
help considerably those maths departments in developing
countries where some research capacities are already
existing or evolving. As a matter of fact, one of the main
obstacles for developing countries to access scientific literature
on a broader scale, are unfavourable exchange
rates. In many cases, however, local currencies cannot
be converted into hard currencies, and that makes matters
even worse for universities in such countries. In one
instance so far, EMS-CDC was able to pay for the subscription
of Zentralblatt using some of the honorariums
which various reviewers of Zentralblatt donate to us. Unfortunately,
our funds are too small to act in this direction
on a regular basis.
As you can see from the above, in widening our scope
of activities we are also faced with the problem of funding
these activities. We fully depend on financial assistance
as mentioned above. In addition, some money came
in directly from national mathematical societies where
individual members donated to EMS-CDC when paying
their membership fees (see the accompanying letter to
Presidents of national mathematical societies below). If
this could happen on a more regular basis, this would be
of great help to us.
At our last annual meeting in April, this year, we had
a regular change of leadership. Tsou Sheung Tsun (Oxford)
took over from Herbert Fleischner the chair of
EMS-CDC (he had been chairman since 2002, and Tsou
Sheung Tsun was the vice-chair). The vice-chair is now
EMS Newsletter December 2007 9

EMS News
Leif Abrahamsson of Uppsala University who has vast
experience related to assisting universities in developing
countries. Herbert Fleischner remains member of EMS-
CDC and will focus his work on assisting in developing
M.Sc. programmes.
However, we agreed with EMS-EC that changes in
the leadership of EMS-CDC, but also changes in membership
should be done gradually since sudden changes
may interrupt activities. We are thus calling upon colleagues
who are interested in participating in EMS-CDC
activities, to contact its leadership. Apart from regular
members we also have some EMS-CDC associates who
are not members as such, but cooperate with EMS-CDC
on a more or less regular basis. Such associates might become
regular EMS-CDC members if they wish to.
More information about our membership, links, collaborations,
activities can be found in a website being
built: http://www.maths.ox.ac.uk/~tsou/ems/.
Tsou Sheung Tsun [tsou@maths.ox.ac.uk], Chair
Leif Abrahamsson [leifab@math.uu.se], Vice-Chair
Herbert Fleischner,[fleisch@dbai.tuwien.ac.at], Programme
Director.
To the Presidents of National
Mathematical Societies
Dear Colleagues,
We are writing in the hope of enlisting your help in a scheme to promote aid to the mathematicians in the developing
world. There is no question that the progress in mathematics in those less privileged parts of the world is
of utmost importance, not only for mathematicians there but also for mathematicians everywhere. And it is also
important that as many mathematicians as possible get engaged in this process.
To this end, a first step is to ask your members to donate a small amount of money, at the time they renew their
membership in your Society, to enable the Committee for Developing Countries of the European Mathematical
Society to carry on with and expand their work. One efficient way we can think of is to provide a space on your
annual renewal form for your members to add an appropriate amount, for example 10 EUR, to their remittance
or cheque for this purpose. Practically speaking, you could ask them to tick a box, or even better, one of two boxes,
with one marked 10 EUR (or local equivalent) and the other blank for them to fill in their desired amount. The
total collected can then be transferred to the EMS-CDC account in Finland, say once a year. We hope this will not
add a significant amount of administrative work to your staff. We are sure that a large number of your members
are happy to make such a donation, if so facilitated by their own Society, as it has already been shown to be the
case in some European national societies already. If yours is one of them, we thank you most heartily and hope
you will continue the good work.
We are planning further appeals to involve more mathematicians in our development work, and not just financially,
because we believe that what works and what counts ultimately is mathematicians helping mathematicians.
After all, it’s mathematicians who care most about mathematics, anywhere in the world.
For information, we are attaching a short description of this committee, which you may wish to use in initiating
the above scheme if you approve it.
Best wishes,
Ari Laptev
(President, European Mathematical Society)
Tsou Sheung Tsun
(Chair, Committee for Developing Countries, EMS)
10 EMS Newsletter December 2007

Summer School in Contemporary
Mathematics: Dubna, Russia, 2007
Michael Eickenberg (Luxembourg)
News
Since 2001, the Russian Academy of Sciences (mathematical
section), the Steklov Mathematical Institute, the Moscow
Department for Education and the Moscow Center
for Continuous Mathematical Education organize a summer
school at Dubna (125 km north of Moscow) that is
unique in its choice of professors and participants. During
two weeks around a hundred students (from the last two
years of high-school or from the first two years of university)
take part in 70-80 lectures and seminars. In 2007, the
School accommodated foreign students for the first time;
contacts were made through the EMS. The Newsletter
asked one of the participants to report on his experiences.
Additional information on the summer school including
the list of speakers and the titles of lecture series can be
found on the web site http://www.mccme.ru/dubna/eng/ .
Description of the event
This year’s summer school took place 19-30 July 2007 in
an installation for education camps in Ratmino, a village
close to the science town Dubna. Classes were attended
in the facilities on site (featuring a grand auditorium in
the main building, a common room in each of the two
dormitories and the library) on every day except the first.
There were usually four lectures of 74 minutes each day,
two as a block in the morning from 9:00 and two in the
afternoon from 14:30, with a break of around 30 minutes
between two lectures of a block. Occasionally there
would be three or five lectures depending on necessity.
In a given time slot there were either up to four, smaller
lectures in parallel or a single plenary lecture. Some
lectures were stand-alone one-time events whilst other,
more elaborate ones continued over several days in the
same time slot and place.
A generous break was given for lunch and recreational
activities, which could consist of swimming in the river
Dubna, playing sports such as football, volleyball, basketball,
table tennis, badminton and chess, or resting, depending
upon personal preference.
Breakfast, lunch and dinner were served in the canteen,
which was also situated in the main building. The
evenings were free; the sports mentioned above were
offered and tournaments were organized, which culminated
in finals on the last day with a little prize and mention
during the closing ceremony (the author managing
to claim first place in badminton :) ). Furthermore there
were numerous cultural activities, such as a presentation
of the works of Tchaikovsky, movies, a poetry evening
and singing along to (or in my case listening to) guitar
music. Talks on non-mathematical subjects such as genetically
modified organisms were given and the events
of the coup d’état attempt of August 1991 that led to
the fall of the soviet union were recalled by Alexander
Muzykantsky, who was vice-chairman of the Moscow
government at the time. In addition to this, during one
afternoon there was a boat ride along the Volga (as luck
would have it, this was the only day it rained but it was
not hard enough to spoil the fun).
The classes were held by some of the most renowned
Russian and international mathematicians to an audience
of around 100 participants whose education did not
exceed second year university level, most of them having
achieved high ranks in mathematical competitions. The
subjects covered many fields of contemporary and classical
mathematics, from algebra and analysis to geometry,
number theory, game theory and logic. For instance, four
proofs of Gödel’s incompleteness theorem were discussed,
as well as the theory of auctions, arithmetic progressions
in prime numbers, polyhedrons and polytopes,
and the likes of harmonic functions defined on fractals
and discrete holomorphic functions.
Outside of lectures, the professors were always approachable
for any questions or conversation. During
free time one could often spot students and teachers engaged
in discussion.
The school from an international participant’s
point of view
Lecture at Dubna
In total there were five international participants, from
Hungary, Finland, Norway and in my case Germany.
We were warmly welcomed into the community and its
friendly, open and down-to-earth atmosphere after having
been guided through the visa application process
EMS Newsletter December 2007 11

News
Discussion after a lecture
Outdoors coaching
and the arrival by the office of the MCCME in Moscow.
Virtually everybody in the camp had a very high level
of English so communication was no problem. From the
third day on there was one lecture per day given in English,
organized especially for us. In the other lectures, including
the plenary ones, there was a sufficient number
of the organising team or participants capable of giving
a running commentary in English. This helped to capture
the essence of what was being conveyed, yet not every
detail – and when a joke needs explaining (provided it
can be explained) it is usually too late … Naturally, not
being able to read exactly what the professor has written
two blackboards before makes it very difficult to catch
up again once you lose the thread.
Personal note
After the remarkable feat of managing to organize my
arrival within a week of the departure date, due to the
quick responses of Vitaly Arnold and Nadezhda Shikova
of the MCCME to my emails, I knew I was in good hands.
My decision to participate came so late because the student
nominated to go decided not to so I was able to take
her place.
The summer school has shown me the vastness and
also the beauty and ‘music’ of contemporary and classical
mathematics. And the fact that it has invoked profound
changes in the way I think is underlined by stating that
because of my participation in this school I have decided
to change my studies from physics to mathematics at the
University of Luxembourg. My thanks go out to all the
organizers, teachers and participants for a great ten days
in the world of mathematics and to the mathematical society
of Luxembourg for sending me there.
Michael Eickenberg [m.eickenberg@web.
de] is a third year student of mathematics
and physics at the University of Luxembourg.
Currently, he is taking a term
at University Henri Poincaré I in Nancy,
France. He expects to write a thesis for a
bachelor degree in 2008. Apart from his
studies he is an eager reader, is interested in languages and
plays Ultimate Frisbee and the guitar.
Journal of Noncommutative Geometry
ISSN: 1661-6952
2008. Vol. 2, 4 issues. Approx. 500 pages. 17.0 cm x 24.0 cm
Price of subscription, including electronic edition:
198 Euro plus shipping (add 20 Euro for normal delivery).
Other subscriptions on request.
Aims and Scope: The Journal of Noncommutative Geometry covers the noncommutative world in all its aspects. It is devoted to publication of
research articles which represent major advances in the area of noncommutative geometry and its applications to other fields of mathematics and
theoretical physics.
Editors: Paul F. Baum, (USA), Jean Bellissard (USA), Alan Carey (Australia), Ali Chamseddine (Lebanon), Joachim Cuntz (Germany), Michel Dubois-Violette (France), Israel M. Gelfand
(USA), Alexander Goncharov (USA), Nigel Higson (USA), Vaughan F. R. Jones (USA), Mikhail Kapranov (USA), Gennadi Kasparov (USA), Masoud Khalkhali (Canada), Maxim Kontsevich
(France), Dirk Kreimer (France and USA), Giovanni Landi (Italy), Jean-Louis Loday (France), Yuri Manin (Germany), Matilde Marcolli (Germany), Henri Moscovici (USA), Ryszard Nest
(Denmark), Marc A. Rieffel (USA), John Roe (USA), Albert Schwarz (USA), Georges Skandalis (France), Boris Tsygan (USA), Michel Van den Bergh (Belgium), Dan-Virgil Voiculescu
(USA), Raimar Wulkenhaar (Germany), Guoliang Yu (USA).
European Mathematical Society Publishing House
Seminar for Applied Mathematics, ETH-Zentrum FLI C4
Fliederstrasse 23
CH-8092 Zürich, Switzerland
subscriptions@ems-ph.org
www.ems-ph.org
12 EMS Newsletter December 2007

Sudoku puzzles and
how to solve them
Andries E. Brouwer (Eindhoven, the Netherlands)
Feature
For a human solver, backtrack is the last resort. If all attempts
at further progress fail, one can always select an open
square, preferably with only a few possibilities, and try these
possibilities one by one – maybe using pencil and eraser, or
maybe copying the partially filled diagram to several auxiliary
sheets of paper and trying each possibility on a separate
sheet of paper.
For very difficult Sudoku puzzles, this is the fastest way
to solve them, also for humans.
However, one solves puzzles not because the answer is
needed, but for fun, in order to exercise one’s capabilities in
logical reasoning. And solving by backtrack is dull, boring,
mindless, no thinking required, better left to a computer, no
fun at all.
So, Backtrack, or Trial & Error, is taboo. And if it cannot
be avoided one prefers some limited form. Maybe whatever
can be done entirely in one’s head.
Two puzzles – the second one is difficult
Sudoku
A Sudoku puzzle (of ‘classical type’) consists of a 9-by-9 matrix
partitioned into nine 3-by-3 submatrices (‘boxes’). Some
of the entries are given, and the puzzle is to find the remaining
entries, under the condition that the nine rows, the nine
columns, and the nine boxes all contain a permutation of the
symbols of some given alphabet of size 9, usually the digits
1-9, or the letters A-I.
Some mathematicians will claim that since this is a finite
problem, it is trivial. The time needed to solve a Sudoku puzzle
is O(1) – indeed, one can always try the 9 81 possible ways
of filling the grid. But one can still ask for efficient ways of
finding a solution. Or, if one knows the solution already, one
can ask for a sequence of logical arguments one can use to
convince someone else of the fact that this really is the unique
solution.
Backtrack and elegance
It is very easy to write an efficient computer solver. Straightforward
backtrack search suffices, and Knuth’s ‘dancing links’
formulation of the backtrack search for an exact covering problem
takes a few microseconds per puzzle on common hardware
today.
Grading
Most Sudoku puzzles one meets are computer-produced, and
it is necessary to have a reasonable estimate of the difficulty
of these puzzles. To this end one needs computer solvers that
mimic human solvers. Thus, one would also implement the
solving steps described below in a Sudoku solving program,
not in order to find the solution as quickly as possible, but in
order to judge the difficulty of the puzzle, or in order to be able
to give hints to a human player. Such AI-type Sudoku solving
programs tend to be a thousand to a million times slower than
straightforward backtrack.
Generating
Given the backtrack solver, generating Sudoku puzzles is easy:
start with an empty grid, and each time the backtrack solver
says that the solution is not unique throw in one more digit. (If
now there is no solution anymore, try a different digit in the
same place.) To generate a puzzle in this way requires maybe
thirty calls to the backtrack solver, less than a millisecond.
One can polish the puzzle a little by checking that none of the
givens is superfluous.
Afterwards one feeds the puzzles that were generated to
a grader. Maybe half will turn out to be very easy, and most
will be rather easy (‘humanly solvable’). It is very difficult to
generate very difficult puzzles, puzzles that are too difficult
even for very experienced humans.
Solving
Below we sketch a possible approach for a human solver.
The goal is to be efficient. In particular, the boring and timeconsuming
action of writing all possibilities in every empty
EMS Newsletter December 2007 13

Feature
square is postponed as much as possible. On the other hand,
some form of markup helps.
Baby steps
When eight digits in some row or column or box are known,
one can find the last missing digit.
Pair markup
If one checks where a given digit can go in some row or column
or box and finds that there are precisely two possibilities,
then it helps to note this down. That is efficient, one does not
do the same argument over and over again, and helps in further
reasoning.
Exercise: (i) Solve this puzzle using baby steps only. (ii) Show
that if a puzzle can be solved using baby steps only, it has at
most 21 open squares.
Singles
When there is only one place for a given digit in a given row
or column or box, write it there. If there is only one digit that
can go in a given square, write it there.
Thus, a small stroke labeled with a digit and joining two squares
indicates that one of the two squares must have that digit.
Given two pairs for the same digit straddling the same two
rows or columns, the digit involved cannot occur elsewhere in
those rows or columns.
Given two pairs between the same two squares, one concludes
that these two squares only contain the two digits involved
and no other digit.
Left: The 1 in the middle right box must be in the colored square. Right:
The 6 in the top row must be in the colored square.
Baby steps are particularly easy cases of singles. Checking
for singles requires 324 steps. Knuth’s ‘dancing links’ backtracker
will take 324 steps if and only if the puzzle can be
solved by singles only. It is unknown how many open squares
a puzzle can have and be solvable by singles only. There are
examples with 17 givens. It is unknown whether any Sudoku
puzzles exist with 16 givens and a unique solution.
Left: Solve using singles only Right: 16 clues and 2 solutions
Matchings
A sudoku defines 36 matchings (1-1 correspondences) of size
9: between positions and digits given a single row or column
or box, and between rows and columns given a single digit.
For each of these 1-1 correspondences between sets X and Y
of size 9, if we have identified subsets A ⊂ X and B ⊂ Y of the
same size n,1≤ n ≤ 9, such that we know that the partners of
every element of B must be in A, thenA and B are matched,
and nothing else has a partner in A. More explicitly:
(A) If for some set of n positions in a single row, column, or
box there are n digits that can be only at these positions, then
these positions do contain these digits (and no other digits).
14 EMS Newsletter December 2007

Feature
For example,
Exercise Complete this Sudoku. The ‘Sue de Coq’ argument
will be useful.
The subset principle
The three digits 3, 4, 9 in the second row must be in columns
4, 6, 9, so the digit 5 cannot be there.
Let S be a subset of the set of cells of a partially filled Sudoku
diagram, and let for each digit d the number of occurrences of
d in S be at most n d .If∑n d = |S|, then the situation is tight:
each digit d must occur precisely n d times in S. In this case
we can eliminate a digit d from the candidates of any cell C
such that the presence of a d in C would force the number of
d’s in S to be less than n d .
For example,
(B)Ifforsomesetofn digits there are n positions in a single
row, column, or box, that cannot contain any digits other than
these, then these digits must be at those positions (and not
elsewhere in the same row, column, or box).
For example,
Here all possibilities for the fields in column 4 are given. Note
the four colored fields: together, they only have the four possibilities
6, 7, 8, 9. So, these colored fields contain 6, 7, 8, 9 in
some order, and we can remove 6, 7, 8, 9 from the possibilities
of the other fields in that column.
(C) Pick a digit d. If for some set of n rows R there is a set of
n columns C such that all occurrences of d in these rows must
be in one of the columns in C, then the digit d does not occur
in a column in C in a row not in R (and the same with rows
and columns interchanged).
(This argument is called X-wing for n = 2, Swordfish for
n = 3, Jellyfish for n = 4.)
For example,
In the five colored squares, the five digits 2,3,4,5,9 each occur
at most once (since all occurrences of 3,4,5 are in a single box
and all occurrences of 2,5,9 in a single column). Since the
situation is tight, digits 3,4,5 do not occur elsewhere in this
box, and digits 2,5,9 do not occur elsewhere in this column.
More generally, one may remove a candidate for a cell
outside S if its presence would force ∑n d < |S|.
Hinge
The previous subsection used that each cell contains at least
one digit. Conversely, each digit is in at least one cell in any
given row, column or box. For example,
Consider the three (colored) columns 1, 5, 7. The only
possibilities (indicated by an asterisk) for a digit 1 in these
columns occur in rows 2, 4, 9. Therefore, digit 1 cannot occur
outside these columns in these rows.
Consider the above diagram. If cell a has a 1, then cell b does
EMS Newsletter December 2007 15

Feature
not have a 1, and then the 1 in row 2 must be in cell c. But
then the colored area cannot contain a 1, impossible. (So, cell
a has a 5.)
Forcing Chains
Consider propositions (i, j)d ‘cell (i, j) has value d’and(i, j)!d
‘cell (i, j) has a value different from d’. By observing the grid
one finds implications among such propositions.
There are at least three obvious types of such implications.
Let us say that two cells are ‘adjacent’ (or, ‘see each other’)
when they lie in the same row, column or box, so that they
must contain different digits. This gives the first type: If (i, j)
is adjacent to (k,l) then (i, j)d > (k,l)!d where > denotes
implication.
In case (i, j) is adjacent to (k,l) and (k,l) only has the two
possibilities d and e, then (i, j)d > (k,l)e. This is the second
type of implication.
Finally, if some row, column or box has only two possible
positions (i, j) and (k,l) for some digit d, then(i, j)!d >
(k,l)d. This is the third type.
Consider chains of implications. If (i, j)d >... >(i, j)!d
then (i, j)!d.
For example
Uniqueness
A properly formulated Sudoku puzzle has a unique solution.
One can assume that a given puzzle actually is properly formulated,
and use that in the reasoning, to exclude branches
that would not lead to a unique solution.
For example, a grid
can be completed in at least two ways, violating the uniqueness
assumption. That means that this must be avoided, so
that
At least one of the corners of the rectangle is 2.
For example,
Here (8,2)6 > (8,6)8 > (5,6)6 > (5,1)5 > (6,2)6 > (8,2)2
was used to conclude (8,2)2.
For example
Here (8,2)1 > (6,2)!1 > (6,9)1 > (4,7)!1 > (8,7)1 >
(8,2)!1 was used to conclude (8,2)!1. For such chains that
involve a single digit only, and where the implication types
alternate between I and III, one often uses a simplified notation
like 1 : (8,2) − (6,2) =(6,9) − (4,7) =(8,7) − (8,2)
where − denotes that at most one is true and = that at least
one is true.
Finding useful chains may be nontrivial, and there are various
techniques such as ‘colouring’ that help.
Look at the colored rectangle. If (7,7)3, then (7,5)8 and
(8,7)8sothat(8,5)3 and we have a forbidden rectangle with
pattern 83–38. So, (7,7)!3, which means that we have an X-
wing: digit 3 in columns 2,7 can only be in rows 8,9 and does
not occur elsewhere in these rows. In particular (9,4)!3 so
that (6,4)3, and (8,5)!3 so that (8,5)8.
More generally:
Theorem. Suppose one writes some (more than 0) candidate
numbers in some of the initially open cells of a given Sudoku
diagram, 0 or 2 in each cell, such that each value occurs 0
or 2 times in any row, column, or box. Then this Sudoku diagram
has an even number of completions that agree with at
least one of the candidates in each cell where candidates were
16 EMS Newsletter December 2007

Feature
given. In particular, if the Sudoku diagram has a unique solution,
then that unique solution differs from both candidates in
at least one cell.
Digit patterns and jigsaw puzzles
A more global approach was described by .
Solve a puzzle until no further progress is made. Then, for
each of the nine digits, write down all possible solution patterns
for that digit. One hopes to find not more than a few
dozen patterns in all. Now the actual solution has one pattern
for each digit, where these 9 patterns partition the grid.
Regard each digit pattern (‘jigsaw piece’) as a boolean formula
(‘this pattern occurs in the solution’). Write down the
formulas that express that for each of the nine digits exactly
one pattern occurs, and that overlapping pieces cannot both
be true. Solve the resulting system of propositional formulas.
This approach allows one to solve some otherwise unapproachable
puzzles.
new to oxford
journals in
2007
INTERNATIONAL MATHEMATICS
RESEARCH NOTICES
www.imrn.oxfordjournals.org
Bibliography
[1] There is an enormous amount of literature on Sudoku on the
web. This note is a condensed version of http://homepages.cwi.
nl/~aeb/games/sudoku/.
This article has previously appeared in Nieuw Archief voor
Wiskunde (5) 7, no. 4, pp. 258–263, December 2006. We
thank the editor for the permission to reproduce it in the Newsletter.
Andries E. Brouwer [aeb@cwi.nl/aeb@win.
tue.nl] is a professor of mathematics at Technical
University of Eindhoven, the Netherlands.
He likes to play games.
INTERNATIONAL MATHEMATICS
RESEARCH PAPERS
www.imrp.oxfordjournals.org
INTERNATIONAL MATHEMATICS
RESEARCH SURVEYS
www.imrs.oxfordjournals.org
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EMS Newsletter December 2007 17

Paulette Libermann, 1919–2007
Obituary
Michèle Audin (Strasbourg, France)
Paulette Libermann
(© Mme Mounier-Veil)
Born on 14 November 1919, Paulette
Libermann died on 10 July 2007.
In 1938, she joined the École Normale
Supérieure de Jeunes Filles, also
called the École de Sèvres. Until then,
this school had only prepared its students
to pass the (feminine) ‘agrégation’,
which would allow them to become
teachers in secondary schools
(for girls). The director Eugénie Cotton
(both a physicist and a left-wing
militant) decided to raise the level of her students to make
them attain the same level as the men of the École Normale
Supérieure. Among the professors who taught these young ladies
were Élie Cartan and two younger mathematicians Lichnerowicz
and Jacqueline Ferrand.
At this point, Paulette Libermann’s story meets History. In
the fall of 1940, she was beginning to prepare for the agrégation
when the so-called French State, anticipating the desire of
German occupying forces, established a series of laws called the
‘status of Jews’. These laws forbid the people they defined as
Jews (among them Paulette Libermann) from practising a certain
number of professions, one of which was teaching. Therefore
she could not pass the aggregation … and this led to Élie
Cartan suggesting that she start research instead.
Later, she said that the anti-Semitic laws were a bit of luck
for her. But at the time, the threat to French Jews was becoming
more and more serious and in 1942 Paulette Libermann’s
family left Paris for Lyon to live a semi-clandestine life until
the liberation.
She then went back to the École de Sèvres to pass the agrégation.
As this was the case for all young French mathematicians
(both men and women) she was sent to teach in a secondary
school. But now she knew that she wanted to do mathematical
research. At this point Élie Cartan gave her a second piece of
good advice, namely to start a thesis with Ehresmann.
Student of Ehresmann
It is hard to imagine what Ehresmann’s school of differential
geometry and topology was at that time. Charles Ehresmann
himself, a member of Bourbali, passed a thesis with Élie Cartan
in 1934. He had many students. The first one, Jacques Feldbau,
proved in 1939 that a bundle over a simplex is a product before
inventing, jointly with Ehresmann, the notion of an associate
bundle … and the exact homotopy sequence of a fibration.
Regrettably, Feldbau was less lucky than Paulette Libermann;
the anti-Semitic policy sent him to his death in a concentration
camp. Georges Reeb (1920–1993), Wu Wen-Tsun, Paulette
Libermann, André Haefliger, Valentin Poenaru, are among the
best known of Ehresmann’s students.
Equivalence problems
Paulette Libermann published numerous papers on differential
geometry. She defended her thesis, Sur le problème d’équivalence
de certaines structures infinitésimales, in 1953. The equivalence
problem is a general problem and has been investigated by
many including Élie Cartan. Roughly speaking, the question is
to classify, up to local isomorphism, structures on manifolds. For
instance, all the manifolds of the same dimension are equivalent
(this is a local question). But this is not true anymore if the manifolds
are endowed with Riemannian metrics: a (curved) sphere
is not locally isometric to a (flat) plane.
The metric can be replaced by a lot of structures, for instance
a family of 1-forms (a Pfaff system), a notion on which Paulette
Libermann worked a lot.
Symplectic geometry
Her results on symplectic geometry have become classics in the 70s
and the 80s (when symplectic geometry became fashionable). The
symplectic equivalence problem is solved by the Darboux theorem:
all symplectic manifolds of the same dimension are locally
isomorphic. There is no local invariant in symplectic geometry!
So:
(1) Look for global invariants. One of the most powerful tools
is Gromov’s theory of pseudo-holomorphic curves (1985).
This is based on the notion of almost complex structures
on symplectic manifolds, one of the notions Paulette Libermann
investigated in her thesis.
(2) Make the structure more rigid, for instance by considering
two transverse Lagrangian foliations on the manifold. This
problem, that she investigated in one of her papers, lies at
the basis of the theory of integrable systems.
She will also be remembered as the author, jointly with Charles-
Michel Marle, of one of the very first textbooks on symplectic
geometry.
After her thesis, she became a professor in Rennes and then
in Paris. She was very helpful to young mathematicians.
Tiny, energetic, smiling, chatty, sometimes caustic, she was
also a memory of the mathematical community. She liked to
speak of those who helped her, either personally or professionally:
Cartan’s family, Jacqueline Ferrand, Ehresmann, anonymous
others … and she also liked to speak of those who did not
help her. She participated, almost until the end, to conferences
all around the world. The last time we met, in April, I was leaving
for Vietnam. I cannot go, she told me, too tiring, I am getting
old. She was 87.
We shall miss her.
Michèle Audin [maudin@math.u-strasbg.
fr] is a professor of mathematics at the Institut
de Recherche Mathématique Avancée,
Strasbourg, France.
EMS Newsletter December 2007 19

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013390x

Leonhard Euler in Berlin
History
Jochen Brüning (Humboldt-Universität Berlin, Germany)
1. Introduction
Three hundred years ago, on 15 April 1707, Leonhard
Euler was born in Basel. Thus, 2007 became a veritable
Euler year (even without being named so by an international
agency) and his achievements have been exposed,
scrutinised and celebrated all around the world. Euler’s
work in mathematics is very likely the most voluminous
and the most influential of any mathematician, not only
for the depth and the variety of his results but also for
his influence on how mathematics is written and taught.
As an indirect proof of this statement we can regard the
fact that a comprehensive scientific biography of Leonhard
Euler is still missing, in spite of the enormous
amount of work devoted to special aspects of his life and
his production.
The remarkable stability of his living conditions
makes it easy to give a rough sketch of Euler’s biography.
Even though he was apparently not a prodigy, his
enormous talents showed early. The son of a protestant
priest, he took up university studies in his home town
Basel at the age of 13, enrolling in various faculties. Euler
studied theology, philology and history, only afterwards
turning to his real favourites, the sciences and above all
mathematics, finishing off even with some physiology.
In 1727 he wrote his dissertation in physics, presenting
a theory of sound on the basis of which he applied for
an open physics professorship in Basel. Since this application
was unsuccessful, the young Euler decided to
follow an invitation to the Petersburg academy in the
same year, solicited by the two sons of Johann Bernoulli
with whom he had studied and who where already there.
Euler began to work in St. Petersburg with the medical
department, gradually shifting his subject towards
astronomy, geography and mathematics, while improving
his standing, and his salary, in the academy. Eventually,
he became a professor of mathematics in 1737. Two
important private events must be mentioned: in 1733,
Euler married his Swiss compatriot Katharina Gsell, the
daughter of a well-known painter and in 1738 he lost his
right eye, probably as the consequence of an infection.
After the death of Peter the Great the political circumstances
in Russia became more and more unstable,
with increasingly unsafe conditions and imminent danger
to the existence of the academy. Euler and his wife were
very disquieted by this. Thus he followed, in 1741, the
prestigious offer by Friedrich II, the new king of Prussia,
to come to Berlin and help construct a new Royal Academy
of Sciences. He arrived in Berlin on 25 July 1741
and he was to stay until 1766. In this year, after getting
more and more disappointed with the king’s handling
of the academy and Euler himself, he followed a second
call to Russia, this time from Katharina II; he returned
Fig. 1: Euler at the age of 30, the only existing portrait showing him
with full eyesight; after an original by Brucker.
© Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften,
Fotosammlung, Leonhard Euler, Nr. 1
to St. Petersburg and stayed there until his death on 18
September 1783.
We devote these pages to the Berlin period in Euler’s
life, trying to highlight not only, and not even primarily,
his scientific work but also his many other activities. We
hope that in this way a more complete picture of Euler
may arise and, perhaps, a better appreciation of how
much we owe to him.
2. People
Even for a person with the mental powers of Euler,
projects and activities are always connected with people.
In fact, Euler was not shy or even introvert; on the contrary,
he enjoyed company for conversations, games and
music, and he loved to appreciate and being appreciated.
EMS Newsletter December 2007 21

History
Fig. 2: Friedrich II, king of Prussia, contemporary engraving; Universitätsbibliothek
Basel. In: EO IV, 6, p. 276
Hence it is helpful to take a closer look at the two persons
who were, without any doubt, most important to Euler’s
everyday life in Berlin, even though he met them in person
only rarely: the king Friedrich II and the president of
the academy Pierre-Louis Moreau de Maupertuis.
Friedrich II
Friedrich II was certainly the most intellectually gifted
monarch of his time but he was also a very complicated
character. All his life he felt attached to philosophy, music
and poetry, and above all to French culture, while he
disliked military exercises and quite possibly the military
as such. His father, Friedrich Wilhelm I, had equipped
Prussia with too large an army but a solid financial basis,
at the expense of a stern and parsimonious regime. Friedrich
suffered a lot in this atmosphere and tried to flee
the country, together with his friend Hans Hermann von
Katte; he was caught, his friend was executed and he was
kept as a prisoner by his father for two years. In 1740,
at the age of 28, Friedrich inherited the throne and was
suddenly confronted with the realities of power, without
much preparation; in fact, he had spent the previous
years in the somewhat remote castle of Rheinsberg, with
a circle of friends enjoying French culture and the arts.
Two decisions of his first weeks in power highlight
the very different aims he now wanted to unite: firstly he
started an unjustified war for the rich province of Silesia
belonging to the Habsburg Empire and secondly he tried
to rebuild the Société des Sciences, the precursor of the
Prussian Academy of Sciences, which had been founded
by Leibniz but had deteriorated very much during the
reign of Friedrich Wilhelm I. The latter decision is important
for our context because it brought Euler to Berlin
in 1741. The first decision was important too as it would
keep Friedrich busy until 1763, the end of the Seven
Years War (incidentally the first war that was really global
in the sense that it was fought on several continents).
Necessarily, the extended warfare crucially limited the
king’s potential of caring for the sciences and the arts,
with the exception of a short period between 1745 and
1753. This is also the period during which Euler felt most
happy being in Berlin, while after 1763 he felt more and
more alienated and thought about going back to Russia,
an idea that became a reality in 1766.
Euler liked Petersburg and the people he met there
very much and he managed to keep very close relations
with his colleagues in the academy and with the Russian
government, even after he had left for Berlin. That
he left at all to accept the offer by Friedrich to come to
Berlin and to reorganise the old Société des Sciences by
winning other prominent scientists and mathematicians
to Berlin is probably due to the political instabilities and
the ensuing uncertainty of daily life after the death of
Peter I, which was especially oppressive for Euler’s wife
Katharina. But once he had accepted the call to Berlin,
he was enthusiastic about the possibilities to organise a
new academy that, according to the goal set by Friedrich,
should be soon among the leading institutions of this
kind in the world. Euler was apparently under the impression
that Friedrich would entrust him with the whole
architecture of this new academy and also with shaping
its scientific, organisational and administrative details.
This, however, was not so: even before calling on
Euler, Friedrich had invited Pierre-Louis Moreau de
Maupertuis to become president of the new academy.
He promised to equip the new president with full powers,
limited only by the king himself, and Maupertuis had
immediately reacted positively to this offer, though he
only took office in 1746. Even then and until his resignation
in 1756, Maupertuis was not continuously present in
Berlin; his stays were interrupted by long periods of absence.
Thus Euler was in fact running the academy in all
practical terms while formally he was only director of the
mathematical class. But he did not even become president
when Maupertuis resigned, which came as a bitter
surprise to him. He had apparently misjudged from the
beginning his relationship with the king, regarding himself
as leading scientific counsellor. For the king, however,
there was a clear difference between a philosophical
approach to government, which he claimed for himself,
and the involvement of scientific advice and judgment in
political decisions. Though Friedrich was fully aware of
the role of technology for the welfare of states and their
political standing, he did not regard scientists in general
as being able to speak out in political matters and to discuss
politics; he certainly appreciated their usefulness but
he wanted to keep them at a distance from the centre of
decisions (he once wrote to Voltaire that the king should
keep a scientific academy as a rustic squire keeps a pack
of hounds, a statement of somewhat exaggerated poign-
22 EMS Newsletter December 2007

History
ancy that nevertheless seems to reveal a deep conviction
often found in politics even today).
The philosopher Voltaire, a leading protagonist of
French enlightenment, was in contact with Friedrich
some time before his enthronement, at the occasion
of which he rejoiced that the sciences and the arts had
come to power. He visited Berlin in 1740 and stayed
there from 1750 to 1753, shaping to a large extent the
style of conversation cultivated in Friedrich’s environment.
Leonhard Euler certainly did not fit the ensuing
ideal of a courtier: his French was not brilliant, his language
was free of irony and cynical jokes and his way of
arguing showed the influence of Calvinist sermons and of
mathematical reasoning. Besides, unfair as it may seem,
Friedrich disliked his one-eyed appearance, calling him
a ‘Cyclops’ in private conversations. Euler certainly felt
this fundamental difference but tried, in the tradition of
the enlightenment, to convince Friedrich of his abilities
and his efficiency by showing that mathematical thinking
and its practical applications provided the best means
to further all interests of the state and the government.
It seems that he worked hard to reach this goal. For example,
immediately after arrival in Berlin he wrote an
essay entitled “Commentatio de matheseos sublimioris
utilitate 1 ”, which was unfortunately not published until
1847. A second example is his book on artillery on which
we will comment below.
This approach conformed very well with Euler’s firm
Calvinist faith, since he believed that God reveals some
of the secrets of the creation through mathematics, allowing
in this way for man-made improvements of the
human condition. But propagating this, Euler confronted
the anti-religious tendency of the French enlightenment
and thus of the Prussian court. Only when he finally understood
that he was not able to bridge this discrepancy,
in spite of all achievements harvested through his singular
abilities, he left Prussia for St. Petersburg.
Pierre-Louis Moreau de Maupertuis
Pierre-Louis Moreau de Maupertuis was a celebrity
when he was asked to become president of Friedrich’s
academy. He had a solid mathematical education that he
had acquired in England and in Basel with the Bernoullis.
He was among the first scientists on the continent
who adopted and popularised Newton’s theories but he
quickly got involved in the attempts to clarify Newton’s
prediction about the flattening of the Earth near the
poles, quite contrary to the view the Cartesians held. After
having been employed as a geometer by the French
Academy in 1731, Maupertuis worked hard for securing
funds to measure a meridian in Lapland, thus being able
to prove or disprove Newton’s assertions. He was successful
in 1736 and he returned from this strenuous expedition
with convincing data that established Newton’s
victory over Descartes and made Maupertuis the most
famous scientist of his time. But his health had suffered
greatly from the strains of the expedition, another reason
for his many absences from meetings of the academy,
and his frequent voyages to France, especially during the
German winter, which was very hard on him.
Fig. 3: Pierre-Louis Moreau de Maupertuis, engraving after the famous
polar expedition of 1736; Universitätsbibliothek Basel.
In: EO IV, 6, p. 29
Even though Maupertuis’ career as a productive scientist
had already been terminated in 1732, he started his
presidency in Berlin with a spectacular result, namely his
“principle of least action”, which was published in 1746. In
this work, he asserted in vague terms, and without making
any use of the new infinitesimal calculus, that the quantity
of “action” he had introduced was minimised in any dynamical
transition of physical states. He based this very
general assertion on metaphysical reasoning, which should
explain the effectiveness, if not parsimony, of the Creator.
Euler could have claimed priority in this matter
since he had published extensively on a general theory
of curves enjoying maximising or minimising properties
with respect to certain functionals given by integrals. This
work began in 1736 and culminated in the book entitled
“Methodus inveniendi lineas curvas maximi minimive
proprietate gaudentes 2 ” published in 1744. This book
presented the first systematic treatment of the calculus
of variations and was described by Constantin Carathéodory
as “the most beautiful mathematical book ever
written”. Quite obviously, Euler’s mathematical treatment
was far superior to Maupertuis’ reasoning and, in
1
“Commentary on the usefulness of higher mathematics”, see
E 790. Here and below we refer in this way to the numbers in
the bibliography of Gustaf Eneström: Verzeichnis der Schriften
Leonhard Eulers in Jahresbericht der Deutschen Mathematiker-Vereinigung,
1. Ergänzung zum 4. Band 1910. Likewise
we will refer to Euler’s Collected Works which appear with
Birkhäuser, Basel, in 4 series with a total of 72 volumes, as e.g.
EO IV, 6 for “Volume 6 of the fourth series”.
2
“A method to invent curved lines with minimal or maximal
properties”, see E 65.
EMS Newsletter December 2007 23

History
particular, he had emphasised from the beginning that it
is not always a question of minimising but that maxima
may occur in nature (which are described in the same
way). However, he did not object to Maupertuis’ statement
that his all embracing principle of least action had
been nicely illustrated by Euler’s work. Euler even explained
at some length that the principle of Maupertuis
was not a mathematical but a philosophical statement
and hence had no place in the world of mathematics.
This statement was certainly compatible with Euler’s
basic convictions but it also showed a certain respect for
the powers that be. After all, Friedrich had bestowed
upon the president of the academy the full power to decide
everything whatsoever and Euler always respected
this fundamental rule. Only this careful respect for the
president’s position made it possible for him to deal with
each and every matter and to carry out in effect what
Maupertuis ordered or might have ordered. The rather
substantial, and extant, exchange of letters between Maupertuis
and Euler, which is explained by the frequent absence
of the president from the academy shows that both
men were, if not outright friends, at least very smoothly
cooperating partners entertaining a style of mutual understanding
and respect. Maupertuis was a politician
and a courtier (where Euler was not) but he was also a
gifted and educated scientist who could understand what
Euler thought and intended. The subtle balance in their
relationship was quite vital for the very positive development
of the Berlin academy from 1741 to 1752.
Family and friends
Saying that Euler was definitely not a courtier does not
mean at all that he was not a person who loved company
and was not able to brilliantly entertain his guests. Nor
does it imply that Euler was unable to keep company
or even friendship with members of the higher society.
To the contrary, his big house in what is today Behrenstraße
not only had enough room for his large family but
regularly housed visitors from abroad. Euler was surely
a family man who has been reported to work at his desk
amidst his many children 3 , without any sign of impatience.
Whatever he did he could interrupt at any time,
taking up a completely different matter and returning,
seemingly uninterrupted, to his concentrated work afterwards.
Besides his nice house, Euler owned an orchard and
a farm (which was recently identified by Wolfgang Knobloch,
see figure 4); on the farm his mother lived as a
widow until her death.
Among the many friends of Leonhard Euler was the
Swiss medallist Johann Carl Hedlinger (1691–1771). The
two compatriots had met in Russia, where Hedlinger had
done some work for Tsareva Elisabeth and her court. He
was then employed by the Swedish king and generally
known as one of the most brilliant medallists in the world.
3
Euler had 13 children with Katharina of whom only 5 reached
adulthood, and only his 3 sons Johann Albrecht (1734–1800),
Karl Johann (1740–1790), and Christoph (1743–1808) survived
him.
Fig. 4: Euler’s farm in Lietzow now Charlottenburg, a part of Berlin;
the Euler possession is marked with the letter “B”.
In: W. Gundlach: Geschichte der Stadt Charlottenburg, 1905,
Beilage XII
Fig. 5: Price medal of the Société des Sciences et des Belles Lettres,
Medailleur Johann Carl Hedlinger, Berlin 1747/48, Ø 67 mm.
© Staatliche Museen zu Berlin-Preußischer Kulturbesitz, Münzkabinett
His fame had also reached the Prussian court and Friedrich’s
counsellor in all matters of architecture and art
Georg Wenzelslaus von Knobelsdorff had inquired under
what conditions Hedlinger would be willing to come
to Berlin. But this letter remained unanswered since Hedlinger
was in Switzerland and Euler stepped in and invited
him and his wife to his home for an extended visit.
From all we know, Hedlinger’s stay in Euler’s house must
have been very pleasant except for the fact that Friedrich
II did not care at all for him, apparently angered by the
negligence shown by Hedlinger in reaction to his generous
offer. After Hedlinger had already spent six months
in Berlin, Friedrich wrote a letter to Euler saying that
he would like to employ Hedlinger, either permanently
or for a couple of years, at any salary Hedlinger might
request.
Now, however, it was too late. Hedlinger was already
resolved to go back to Sweden from where impatient signals
were coming asking for his return. But as much as
Hedlinger hated the style of conversation at the Prussian
court, like his friend Euler he admired Friedrich as the
24 EMS Newsletter December 2007

History
brilliant monarch of the enlightenment and he had long
wanted to produce an image of him. Thus he had made
preparations during his stay but only much later did the
work come to perfection. In 1748 he produced a medal
for the academy (see figure 5), based on discussions with
both Euler and Maupertuis, who were so impressed that
they elevated Hedlinger to the state of honorary member
of the academy. In 1750 he completed the medal for
Friedrich II. The king was delighted with this work and
wanted to buy the stamp at any cost, as he wrote to Euler,
but soon afterwards he did not remember that he wanted
to pay anything to Hedlinger; in spite of all his talents
and all his brilliance, Friedrich II remained the son of his
father in trying to save money wherever he could. Many
projects were damaged or even prevented by his miserly
attitude.
3. Projects
Fig. 6: Sketch of the seating arrangement for the opening ceremony of
the new academy, January 24, 1744.
© Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften,
Bestand Preußische Akademie der Wissenschaften, I-J-S, Bl. 153-154
The Academy reform
As mentioned before, the academy in Berlin was founded
by Gottfried Wilhelm Leibniz in the year 1700 under
the name Société des Sciences. After a slow development
in the first decade of its existence the academy was finally
opened officially on 19 January 1711. It resided in
a building Unter den Linden, known as “Der neue Marstall”
where the older royal academy of arts had found its
quarters in the year 1700. This cohabitation of scholars
and artists and their usually quite different institutions
was unique in Europe at that time. The alimentation of
the academy was effected through the so-called “Kalender-Privileg”,
that is the monopoly to produce calendars
and related publications throughout Prussia. In this way
a reliable though narrow financial basis was provided,
which nevertheless did not allow the realisation of the
ambitious dreams of Leibniz. In addition, the main burden
caused by the bread-and-butter work of calendarmaking
fell upon the mathematicians and astronomers
in the academy, a fact that regularly gave rise to some
disturbances. In addition, during the reign of Friedrich
Wilhelm I (1713–1740) the Société des Sciences deteriorated
under the total neglect if not contempt shown by
the king, who installed his jester as its president. Nevertheless,
some outstanding scholars were working in Berlin
even then, like the philologist and botanist Johann
Leonhard Frisch.
As mentioned above, Friedrich II was prepared to
change things as soon as he came to power, an intention
that was realised with the calls he sent out to some of
the most prominent scientists of Europe preferring, of
course, those conforming with his ideal, the philosophical
attitude of French persuasion. When Friedrich II
called Euler to Berlin he probably did not know of his
enormous energies extending to everything in his neighbourhood
that offered possibilities for an effective treatment.
Thus Euler started immediately to reorganise the
academy, being instructed by the envoys of the king that
he was aiming at an institution only to be rivalled by the
Paris and London academies. Euler’s patience was certainly
severely challenged by the first two Silesian wars,
which absorbed all the energy of the government and
left little room for matters concerning the academy. But
Euler worked relentlessly, relying on his Swiss tenacity,
and certainly not without effect.
The long interrupted series of academy publications
came to new life, the first volume being full of publications
authored by Euler, like many others to come. He
insisted that meetings of the academy would happen under
a regular schedule and with substantial protocols. He
brought new people to Berlin and constantly developed
his communication network, which included most of the
significant scientists of his time. Fortunately he was not
alone and some influential people in the neighbourhood
of the king also wanted progress in academy matters,
even though their emphasis was more on arts and literature.
In 1743, a Société de Belles Lettres was founded
as a further precursor of the promised new academy.
Honouring these attempts, Leonhard Euler worked on
a unification of the two societies and was successful on
24 January 1744 when the new Académie Royale des Sciences
et des Belles Lettres was founded in the Berliner
Stadtschloss.
The organisation was entirely Euler’s, from the conception
of the statutes to the details of the opening ceremony,
and it shows some remarkable aspects. Notably,
among the four classes representing the sciences and
the humanities, the Classe de Philosophies was unique
in Europe, attesting to the important role this academy
would play soon in the philosophical disputes and quarrels
of the 18 th century. As we can see from figure 6, Euler
already appears as director of the mathematical class,
in which function he was officially installed only on 03
March 1746. Thus, in spite of all shortcomings and frustrations,
Euler quickly built a sound basis for what was
to become probably the most fruitful period of his life.
EMS Newsletter December 2007 25

History
Fig. 7: Academy building Unter den Linden, its location from 1752 to
1904; drawing by Calau, engraving by Lauréns and Thiele.
© Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften,
Grafiksammlung, P/BON – 1135
In 1752, after a fire had destroyed the “Neue Marstall”,
both academies moved into a new and quite representative
building, which had to make room for the new Prussian
state library only in 1903 (figure 7).
Practical mathematics
As we have noted already, Euler was resolved to prove
the power of pure mathematics through practical applications,
not only because he wanted to convince his king
of the possibilities of his own field, and hence of his own
abilities, but also as a matter of principle, since it was his
deepest conviction that it is through mathematics that we
can get insight into the work and even the intentions of
the Creator. Naturally, the academy had to serve the king
and the state in many respects, in engineering projects, in
questions of time measurement and calendar calculation,
and regularly in the evaluation of technological innovations,
acting very much like a modern patent office. Euler
was involved in these activities a lot, including some
greater projects. The first among these was the rebuilding
of the Finow canal, a waterway that provided a direct
connection between Berlin and the river Oder but which
had deteriorated over the years. Euler was a member of
the commission that had to visit the canal, his task being
the exact measurement, the levelling, on which corrections
had to be based. Fortunately, not all his missions
expanded into such strenuous expeditions.
Another well-known story concerns the fountain of
Sanssouci and the dream of the king to have one of tremendous
height such as to impress any visitor. It is usually
told that Euler gave detailed advice that failed to
comply with reality and that Friedrich never saw a fountain
higher than 30 centimetres. This outcome in turn
strengthened Friedrich’s contempt of mathematics and
mathematicians. The latter statement is true but some
of the facts have to be corrected, as was carefully shown
by M. Eckert 4 . His detailed account shows that Euler’s
analysis was quite appropriate and that his calculations
would have given the desired result if only the stinginess
of king Friedrich had not interfered again. Thus he twice
selected completely inexperienced but cheap craftsmen
and ordered them to use wood for the water conducting
tubes, a material which simply could not resist the necessary
pressure, while Euler had repeatedly insisted upon
tubes made from lead.
Other tasks of Euler were closer to mathematics, for
example when the king enquired about the correct setup
of a lottery in order to fill the notoriously empty coffers
of the state. The first such lottery had been established in
Berlin in 1740 but it did not live up to expectations. Euler
was also asked to think about the basis of a life insurance
system and a (restricted) pension system, ideas which attest
to the modernity of Friedrich’s thinking.
Another project with practical implications was
closer to Euler’s mathematical thinking than the aforementioned
ones and seems to have played a particular
role in his attempts to convince the king of the usefulness
of mathematics. Finding the Prussian army involved in
extended warfare, Euler must have thought hard about
an application of mathematics in this context, especially
one that would improve the performance of the Prussian
weapons in an undisputable manner. Knowing very well
that successful applications on a larger scale are impossible
without calibrating experiments, it must have come
like a heavenly gift to Euler when he discovered the
book “New Principles of Gunnery”, which was printed
in London in 1742. The author Benjamin Robins was not
unknown to Euler since he owed to him a very negative
criticism of his first book on mechanics written in
St. Petersburg. However, like in the case of Maupertuis’
priority, Euler cared very little about such questions and
did not retaliate in his treatment of Robins’ book, even
though he found many mistakes that had to be corrected.
In this way, he improved the work greatly and elevated
Robins to a fame that he would never have achieved otherwise.
At any rate, what Euler was interested in were the
results of Robins’ experiment; he had found a rather ingenious
way to measure the true velocity of cannon balls
and the force of gunpowder.
With this data, Euler was able to apply infinitesimal
calculus, thus creating the foundations of modern ballistics.
Much to his satisfaction he could improve tremendously
the results and predictions of Robins, correcting
along the way a formula for the air resistance given by
Newton by a factor of two. The new book by Euler finally
consisted of a translation of Robins work enriched with
extensive comments, theoretical developments and calculations,
which resulted in a volume five times the size
of the original, with the appropriate baroque title “Neue
Grundsätze der Artillerie enthaltend die Bestimmung
der Gewalt des Pulvers nebst einer Untersuchung über
den Unterscheid des Wiederstands der Luft in schnellen
und langsamen Bewegungen aus dem Englischen des
4
M. Eckert: Euler and the fountains of Sanssouci. Archive for
the History of the Exact Sciences 56 (2002), 451–468.
26 EMS Newsletter December 2007

History
Herrn Benjamin Robins übersetzt und mit den nöthigen
Erläuterungen und vielen Anmerkungen versehen von
Leonhard Euler königl. Professor in Berlin. Berlin bey
A. Haude königl. und der Academie der Wissenschaften
privil. Buchhändler. 1745” 5 .
Euler certainly met the success he intended, at least
outside Prussia, since his work was immediately introduced
to military schools, notably in France from where
he received significant praise and a sizeable remuneration.
Napoleon, whose love for mathematics is wellknown,
had to study Euler’s book as a young officer.
However, we do not have any proof of a similar reaction
from the Prussian court. We know about a letter that
Euler sent to Friedrich II some time in 1744 wherein he
announces his plan to translate and extend Robins’ book
and asks for permission to devote his time to this project
but we don’t know of any answer. We also know of the
letter of dedication to the king accompanying the completed
book. This letter is dated 20 April 1745 and seems
to be hitherto unpublished.
Euler makes it quite clear, in spite of the formal modesty
of his writing, that he regards this piece of work as
the desired proof of the all embracing power of higher
mathematics. He describes the relationship between what
he calls here “Theoretical Mathematics” and its applications,
which the king seems to have asked for, as adding
experimental data to determine the constants of integration
to the equations of motion that arise from general
principles like his calculus of variations. It seems that the
idea of equivalent but different theories describing the
same phenomena was still alien to Euler, his religious
convictions leading him to believe in “true equations”.
Alas, we do not know of any comments or signs of grace
from the side of Friedrich in reaction to this remarkable
achievement of Euler’s.
Fig. 8: Official list of the damages caused by plunderings in the village
Lietzow 1760.
© Brandenburgisches Landeshauptarchiv, Rep. 2: Kurmärkische
Kriegs- und Domänenkammer, Nr. S 3498
Private Matters
As should be expected, Euler also handled his family
business with great care and great effectiveness. Thus, the
Euler family could be called wealthy since Leonhard not
only enjoyed a rather exceptional salary but also earned
considerable money from many academy prizes he won,
among which we find the prize of the French academy
at least twelve times. Besides, he had revenues from his
farmland, which must have also been significant. At least
we know that during the Russian-Saxonian occupation of
1760 Euler lost, by the official record, 2 horses, 13 cattle, 7
pigs and 12 sheep (see figure 8). For this damage he was
reimbursed by the occupation forces and received, on top
of this, a very generous compensation from Katharina
the Great. This Russian generosity had, unfortunately, no
parallel in Prussia, which made it eventually even easier
for Euler to go back to St. Petersburg. There have probably
been many other sources of occasional monetary
5
New principles of gunnery with determination of the true
force of the gunpowder and an investigation of the different
air resistance for fast and slow motions. From the English of
Mr. Benjamin Robins translated and where necessary commented
on and amended by Leonhard Euler, etc.
Fig. 9: Emanuel Handmann, Portrait of the mathematician Leonhard
Euler; pastel on paper, 57 x 44 cm, 1753.
© Kunstmuseum Basel, Inv. Nr. 276, Foto by M. Bühler
EMS Newsletter December 2007 27

History
Fig. 10: List of books which Leonhard Euler has written or published
while he was in Berlin, 1741–1766.
gains, e.g. a lottery prize, and a careful look at the famous
Handmann portrait (figure 9) reveals that Euler is clad
in silk that was produced from the Berlin academy’s own
mulberry plantation.
4. Scientific work
As mentioned above, a comprehensive treatment of Euler’s
scientific oeuvre is missing in the literature whereas
there is an abundance of detailed discussions of specific
aspects of his work. In the framework of a single essay any
in-depth study is ruled out if one tries to describe some
extended period of Euler’s life, as we do here. However,
it is easy enough to collect some statistics of what he was
doing during his 25 years in Berlin. Thus we can say that
he prepared roughly 380 articles or books in this time of
which 275 were also published in this period 6 . For this,
we rely on the very careful catalogue of Euler’s writings
compiled by Eneström (see footnote 1), who comes up
with a total of 866 (by also counting letters and prefaces
and the like but not reprints or translations). There is no
way of briefly surveying this massive production but the
printed books provide a fairly accurate guide to the areas
of interest to which Euler devoted a substantial portion
of his time in Berlin; a complete list is given in figure 10.
From the rich menu that Euler serves us here, I want to
select two important topics for a somewhat closer scrutiny.
The birth of analysis
Euler’s continuous and widespread flow of work developed
from various sources: from questions that had become
famous because they had withstood the attempts
of many illustrious colleagues, like the Basel problem;
derived from personal interests like the building of
ships 7 ; questions about optics, certainly fostered by the
injuries to his eyes; and problems arising from accidental
impulses, like his work on the foundation of ballistics.
Besides, Leonhard Euler was very familiar with the work
of the giants of the past and eager to develop their work
further or even correct them or prove outstanding conjectures
8 . More importantly, in spite of being constantly
absorbed by considerations directed at the solution of
specific problems, it seems that he always kept in mind
the theoretical framework he was working in and that he
was able to adapt the whole architecture of the relevant
theory according to the new notions and arguments that
protruded from a special study. Whenever he felt the
ideas to have reached a certain maturity, he put them
together in a book and those books usually remained
highly influential for a long time.
The process of development did not stop, though,
once he had written a book and improvements or modifications
he introduced later on in an article often did
not reach the public for quite some time; his educational
style of writing was simply too convincing.
Let us try to exemplify this sketch of Euler’s working
habits with his “analytic trilogy” consisting of the books
marked 1748 A, 1755, and 1768 B in the list of figure 10.
It is a well-known fact that Euler built the foundation for
what we call “Analysis”, that is the theory of real functions,
the various limit processes that build them, and the
differential and integral calculus from which most of the
interesting functions can be derived. What is maybe less
known is the fact that in Euler’s work for the first time
analysis arises on a foundation that is independent of
geometry (while elementary algebra is presupposed and
used as an appropriate substitute). It fits this picture that
Euler, like most other early analysts, was called a “geometer”,
but an even more striking illustration is provided
by the total absence of diagrams, which have accompanied
mathematical texts since the days of Euclid, and a
closer look reveals that this effect is by no means compensated
by an abundance of formulae; in Euler’s texts,
the written language dominates by far.
If we look at the table of contents of the “Introductio
in analysin infinitorum”, we see that the book is divided
into two parts, the first one establishing the foundations
of analysis, the second one building what we would call
linear algebra. That the two parts arise in this order is
6
The publication of scientific articles by Euler was not finished
before 1862 (with Eneström’s number E 856), almost 80 years
after his death.
7
It seems to be unknown why Euler was so much attracted to
ships which he could hardly be familiar with.
8
That we talk about “Fermat’s Last Theorem” today is due to
the fact that Euler had proved, one by one, all other conjectures
left behind by Fermat.
28 EMS Newsletter December 2007

History
another proof of the above statement that, with Euler,
analysis arises independently of geometry. The main goal
of the first part is the study of functions with real arguments
and real values, though occasionally also complex
numbers are considered. Euler does not give a motivation
for this since he could assume that his readers already
knew enough examples for the overwhelming importance
of functions for all applications of mathematics
and had some experience with polynomials and rational
functions. He insists, however (in §4), that a function must
be given by analytic expressions under which notion he
covers the basic algebraic formulae and the specific limit
expressions that are the main topic of the whole book.
Then he proceeds to introduce these limit expressions,
notably infinite series and, in particular, power series and
infinite products; the last chapter is devoted to continuous
fractions.
He then makes systematic use of these operations to
introduce the elementary transcendental functions and,
as a particular highlight, he presents the trigonometric
functions as power series, without any recourse to their
geometric definition. Along the way he shows what powerful
conclusions can be reached by relating infinite series
and infinite product representations of a function to
each other. Transferring the decomposition of polynomials
in linear factors to entire functions he solves the
“Basel problem” mentioned above, which asks for the
precise value of the sum of inverse squares of the positive
integers; in fact, he computes the value of Riemann’s
(perhaps more correctly Euler’s) zeta function at all positive
even integers.
It is fair to say that Euler prefigures here the modern
(physics) concept of a partition function. His arguments,
as is well-known, are not entirely satisfying with respect
to their rigour and some of his conclusions are extremely
bold indeed. It seems, however, that Euler was completely
convinced of the truth of his assertions but did not
want to spoil the flow of the presentation by leaving out
a beautiful result. In addition, he was probably too impatient
to postpone the publication to a later date when he
would have arrived at more convincing proofs. And this
he did in many cases, especially in the case of the Basel
problem for which he provided many more proofs later
and for which he had numerical corroborations when he
first published it.
The second volume in the trilogy concerns differential
calculus (volume “1755” in the list of figure 10), which
builds explicitly on the “Introductio” just discussed. The
preface is a marvellous piece to read (especially in Latin)
and offers some striking features. First of all, Euler immediately
develops a very general notion of function that
could easily be identified with our modern view, freeing
the definition completely from any requirement of analytic
expressions. This change has been little noticed, as
can be seen from the fact that the modern definition of
function is usually attributed to Dirichlet and thus placed
a hundred years later. Next, Euler deals at length with
the problem of “evanescent quantities”, to make it perfectly
clear that a quotient of quantities tending to zero
may have a finite limit. He also indicates here that the
task of determining these limits is of vital importance for
applications since the real problems posed by nature are
only understood by solving differential equations. These
equations are derived, constituted and solved by infinitesimal
procedures only; we would not be surprised if
Euler had added here that “nature is simple only infinitesimally”,
a remark due to Einstein.
A further interesting feature of this preface is the fact
that Euler illustrates the reasoning of this introduction
by just one example, namely the firing of cannon balls
(which he had dealt with at length, as we know). Moreover,
he calls the differential quotient in this connection
the “ratio ultima”, which must have had an ironic connotation
for his contemporaries who knew that Friedrich
II had written “Ultima Ratio Regis”, the last resort of the
king, on his cannons, following the example of Richelieu.
This nice pun may explain why Euler did not illustrate
the goals of the calculus by explaining Newton’s brilliant
derivation of Kepler’s laws.
The fact that the ensuing chapters present the material
of higher differential analysis very much in the way
it is presented in most calculus courses nowadays may
come as a surprise for some but is easily explained by
the fact that Euler’s work was copied ever after, at least
indirectly, notwithstanding many refinements and extensions.
This is also true of the third volume in the analytic
trilogy, which is devoted to the art of integration and will
always remain its true and lasting foundation.
Euler’s architecture of analysis does convince by its
methodological consequence but not so much by its rigour.
It furnishes a very impressive proof of Euler’s analytic
instinct that his edifice lived through many logical
crises only to become reinforced and augmented, without
significantly changing its structure or losing its beauty.
Educational writings
In the thinking of Leonhard Euler we find a remarkable
emphasis on the presentation of the many insights he
had. He always tried to make his thinking clear to other
people and not only to those of comparable insight, as if
the truth of a thought could only be established by communicating
it successfully; here again it seems that we
meet the influence of the Calvinist sermon. Be this as it
may, Euler engaged himself in mathematical education
early on and this occupation accompanied him all his life.
As early as 1735 he wrote a very successful schoolbook
on arithmetic (E 17: Einleitung zur Rechen-Kunst, zum
Gebrauch des Gymnasii bey der Kayserlichen Academie
der Wissenschafften in St. Petersburg. Gedruckt in der
Academischen Buchdruckerey – 1738, 1740).
In Berlin he conceived his educational masterpiece,
a collection of 234 letters to a German princess on questions
of physics and philosophy. These letters were written
to the daughter of the margrave of Brandenburg-
Schwedt with whom Euler kept very friendly relations.
Thus he was asked to give private lessons to the margrave’s
daughter, by then 16 years old, in science and
philosophy. When, in the course of the Seven Years War,
the Prussian court left Berlin temporarily, anticipating
the short occupation of Berlin by Russian and Saxonian
EMS Newsletter December 2007 29

History
troops in 1760, Euler was forced to continue his lessons
by letter. Apparently he was already resolved to combine
these letters into a book, which appeared only much
later, when he was already back in St. Petersburg (E 343,
344, 417 Lettres à une princesse d’Allemagne sur divers
sujets de physique et de philosophie Tome première A
Saint Pétersbourg de l’imprimérie de l’académie impériale
des sciences. 1768, 1769, 1770).
Reading this book is a pleasure even today and one
cannot but admire the clarity and the ease with which
Euler explains very difficult matters, like the constitution
of the solar system according to Newton or the six
possibilities to measure longitude at sea that were in use
or under examination in Euler’s days. We note in passing
that Euler contributed substantially to the eventual
solution of the problem and received a (small) gratification
from the English parliament that had offered a sum
of up to £100,000 for any method that would allow the
measurement of longitude with enough precision; larger
portions of this money were allocated to the clockmaker
Thomas Harris and to the mathematician and geographer
Tobias Mayer who, in turn, had made substantial
use of Euler’s theory in compiling his lunar tables. Other
remarkable parts of Euler’s letters concern an exposition
of elementary logic, where we meet what is now
known as Venn diagrams, Euler’s own theory of light
and sound, which to some extent predated the optical
theories of the 19th century, and his review of the major
philosophical positions of the 18th century and their origins.
That Euler was understandable when writing about
such complicated matters is proved best by the fact that
these letters were among the most economically successful
books of the 18th century and that they have been
reprinted and translated many times in almost 250 years
that have passed since their first appearance; even a
modern reader will enjoy Euler’s remarkable clarity of
exposition even if some parts have inevitably become
obsolete with time.
Another project of pedagogical as well as scientific
nature concerned an atlas designed for the schoolchildren
in Prussia. This enterprise, initiated by the king and
ordered by the president of the academy showed Euler
in all his capacities in an exemplary way. As we would
expect, Maupertuis entrusted Euler with the details of
the project, which comprised the selection, construction
and design of the maps to be shown, the selection, contracting
and paying of the craftsmen who should work
on the project, and finally the printing and the distribution
of the completed work. Scientifically, the resulting
book showed a few innovations, like unusual projections
or careful depictions of the lines of magnetic aberration.
For the practical use of schoolchildren a much smaller
format was chosen differing greatly from the usual atlas
formats, an example setting a new standard quickly. The
book appeared in 1753, followed by a second edition in
1760.
The final masterpiece in Euler’s educational work is
the Vollständige Anleitung zur Algebra von Hrn. Leonhard
Euler. 1. Theil. Von den verschiedenen Rechnungs-
Arten, Verhältnissen und Proportionen. St. Petersburg,
gedruckt bey der Kays. Acad. der Wissenschaften 1770 (E
387, 388). Euler’s pedagogical fame is underlined by the
well-known story (or maybe legend) that the servant to
whom Euler, then completely blind, dictated the manuscript
in St. Petersburg was afterwards quite competent
in algebra although he had never experienced any formal
education. Even though the book was produced in St. Petersburg,
there is little doubt that the main body of the
material had already been outlined in Berlin.
Leonhard Euler was an exceptional human being
and a singular scientist and mathematician, to be compared
only to Archimedes, Newton, and Gauss. His habit
of relentless work, carried out with the greatest care and
indefatigable energy, was based on the firm belief that
the world can be understood and changed for the better
by applying the scientific method of rational explanation.
He applied his seemingly unlimited intellectual
powers to all questions he was confronted with, always
looking for new concepts and fruitful ideas even before
he had a definite method to handle them. His only major
limitation arose from his religious faith, which was at
the same time the major source of his strength, deeply
rooted in his Swiss origin and his Calvinist family background.
Jochen Brüning [bruening@mathematik.
hu-berlin.de] (born 1947) was a mathematics
professor in München, Duisburg
and Augsburg. Since 1995, he has been the
Chair for Mathematical Analysis at Humboldt-Universität
zu Berlin. From 1990 to
1995 he was acting director of the Institute
for European Cultural History (Augsburg). Since 1999, he
has been acting director of the Hermann von Helmholtz-
Zentrum für Kulturtechnik (HU Berlin) and since 2002
he has been a member of the Berlin-Brandenburgische
Akademie der Wissenschaften. His research and publications
are in cultural history and mathematics (especially in
geometric analysis and spectral theory).
30 EMS Newsletter December 2007

An Interview with Beno Eckmann
Interview
Conducted by Martin Raussen (Aalborg, Denmark) and Alain Valette (Neuchâtel, Switzerland)
in Zurich on 10 January 2007.
Education
Professor Eckmann, you were born on 31 March 1917
in Bern, Switzerland, and you are approaching your
90 th birthday now. Could you please tell us a bit about
your school education, in particular who and what
aroused your interest in mathematics?
I went to school in Bern. I will mainly talk about high
school, which is called “gymnasium”. I did the classical
gymnasium – that means with Greek and Latin and languages.
Everything was very good. Except mathematics;
mathematics was very weak! I don’t regret that I studied
Greek and Latin. And I still know Latin well.
I really don’t know why I decided to study mathematics.
It is not that I no longer remember. I just don’t know! I
was thinking about German languages or other languages,
or something else – all kind of things! All of a sudden, I
said: “I want to study mathematics” – here in Zurich, at the
ETH (Swiss Federal Institute of Technology). Someone
told me: “Don’t study mathematics. It’s a very old science.
Everything is known. There’s nothing to get interested
in.” Nevertheless, I went to Zurich and studied maths!
How old were you when you started?
Eighteen years.
What was your student time like in Zurich and who
were your most important teachers and supervisors?
In the first year we had Michel Plancherel (1885–1967),
Ferdinand Gonseth (1890–1975) and as a supervising assistant
Eduard Stiefel (1909–1978). Plancherel was very
old-fashioned, extremely old-fashioned; but in fact he
was not bad! Since I was not really properly prepared,
linear algebra and analysis were quite difficult for me.
However, everything we learned was a revelation and I
realised that mathematics was indeed something I had
expected in my dreams.
The second and later years brought even more interesting
teachers: Heinz Hopf (1894–1971), George Polya
(1887–1985) and Wolfgang Pauli (1900–1958). As we
understood, Hopf was working in a new field: algebraic
topology, a higher type of geometry. I decided early that
later on, I would try to work with him. Hopf was very modern,
he taught in the style of van der Waerden’s “Moderne
Algebra” or later Bourbaki. In fact, at a very early stage,
I started to read B.L. van der Waerdens’ book, “Modern
Algebra” (later called “Algebra”). Mathematical objects
were sets provided with additional structure fulfilling certain
axioms. This was exactly what made the definitions of
Hopf very clear and transparent (groups, spaces, etc.).
Polya was a very good teacher. But he was always
far too slow in the beginning, and in the end the courses
Beno Eckmann during the interview (photos: Indira Lara Chatterji)
were too difficult. Moreover, his definitions were often
not that clear. His books with Szegö were very interesting.
One could learn a lot from the problems when he
followed them chapter by chapter.
As for Pauli, he gave a course in theoretical physics.
Even though I was not really involved in physics, I realised
that in his thinking all types of mathematics were
involved – we had the possibility to get acquainted with
many highly interesting aspects.
How many students were you altogether at the time?
Six students. We were twelve in the whole group: six
mathematicians and six physicists. We were practically
always together; there was not much difference between
us except that the physicists had to go to the laboratories
more often.
You graduated from the ETH at the dawn of World War II.
Yes, I got the diploma in 1939; this corresponds to a Masters
Thesis today. I did my diploma thesis in topology under
Hopf’s supervision. He was really nice and a good
man.
Please tell us about your doctoral thesis work!
After the diploma, I became an assistant to Professor
Walter Saxer (1896–1974). There were not many assistants
at the time because there were not many engineering
students who needed assistants. Saxer was an analyst,
not worldwide renowned but a good professor; he needed
EMS Newsletter December 2007 31

Interview
Beno Eckmann with interviewers Alain Valette (left) and Martin
Raussen (right) on top of the ETH building
assistants because he became rector at the ETH. As his
assistant I replaced him at times and taught the problem
sessions for him. That was a very good training.
Simultaneously I could start working on my PhD
with Hopf. He asked: “Do you want to work on something
else?” But I started immediately on the theme he
gave me, which was homotopy groups. Nobody else had
worked on homotopy groups, except Witold Hurewicz
(1904–1956) in his famous and absolutely wonderful
notes.
Career
Having finished your thesis, you were appointed to
Lausanne.
Indeed, right after the PhD, in 1942. While I was in
Lausanne, I remained lecturing at the ETH. At that
time, I concentrated on combinatorial problems; I do
not know why! At Lausanne, I became acquainted with
Georges de Rham (1903–1990). He lived in Lausanne and
he was a professor at both Lausanne and Geneva. My
main mathematical contacts at the time were with him
and with Hopf.
It was wartime in Europe and you had to serve military
service at that time. What did you have to do?
Of course, I went to the Army, serving in the mountain artillery.
We had to stay in the mountains, normally in summer,
for two weeks at a time. Then I could go back for two
weeks to give all my lectures in Lausanne, and so on.
But there was no communication with abroad during
the war?
Very little. There was some before France was completely
occupied. There was the “free zone” in the south. Charles
Ehresmann (1905–1979) was in the free zone. He came to
Switzerland; we had vacations together.
How did this situation change after the war?
In 1947, I went to the Institute for Advanced Studies (IAS)
in Princeton for an academic year. I had to get myself to
learn English at first, since I was supposed to give lectures
in English, like everybody! Very few of my colleagues had
a good knowledge of English; it was not part of the school
curriculum everywhere. It was particularly important for
my mathematical development that I had the opportunity
to meet people like Solomon Lefschetz (1884–1972) and
Norman Steenrod (1910–1971) at the IAS.
One year after that year in Princeton, in 1948, I was
appointed at ETH. Soon after, Robert Oppenheimer
(1904–1967) became the director of IAS. He invited me
to spend another year at IAS, from 1951 to 1952. Again,
I met interesting people, among them Raoul Bott (1926–
2005) who was then a beginner with fascinating ideas. I
had the possibility to discuss many different topics with
Albert Einstein (1879–1955); he was happy to talk German
and to remember his old experiences from Switzerland.
You travelled a lot to the United States and to other
countries.
I went regularly to the US. Not for the full academic year
but for summer vacation or shorter periods. MSRI at
Berkeley was established and I went there when it was
still very young and talked a great deal to Shiing–Chen
Chern (1911–2004). He explained that they planned to
have a specific topic for every year and they would invite
people for that year. But it never worked that way:
people would come for some period and then they would
perhaps come two years later, and so on.
Scientific work
Under the influence of Heinz Hopf, you started to work
in homotopy theory at a time when algebraic topology
was hardly established. Please give us some reminiscences
on the development. You must be one of the few
left who can witness that algebraic topology has not
always been associated with commutative diagrams,
exact sequences, spectral sequences and so on.
Yes, indeed – exact sequences, commutative diagrams.
When I wrote my first paper they did not exist at all. Not
even a map was denoted as we do it today, with an arrow
from its domain to its codomain. It’s unbelievable! It was
much more difficult to express things and to compute the
exactness of a sequence. At each stage you had to show
explicitly what you needed. And then, as soon as maps
were denoted just by two letters with an arrow in-between
and with this a suitable notation for exact sequences and
diagrams, everything became simple and clear, and you
could use them for clear statements and easy proofs. So
many things that we used a lot of energy on in the past are
almost obvious today!
And then you had to bring in homological algebra...
You see, if a topological space X is acyclic and has fundamental
group G, then it follows from Hurewicz theory
that the homology of X depends on G alone. Thus we
were looking for an algebraic description of the homology
depending only on G. It makes use of the group algebra
of G, and thus the homology of a group algebra was
introduced. These were problems that many people were
32 EMS Newsletter December 2007

Interview
dealing with but it seems not to be widely known that
Heinz Hopf was the first person to construct a free resolution
over a group ring. People don’t know that; they
talk about Eilenberg-McLane 1 but it was Heinz Hopf
who invented free resolutions. He also phrased precisely
what it means that two free resolutions are equivalent. I
carried this line of thought further on.
Let us talk about your many other contributions to
mathematics. Apart from algebraic topology, your
name is associated with results in differential geometry,
group theory and more recently L 2 -invariants, at the
boundary between topology and analysis. How would
you describe the common thread through your work?
Topology, in the spirit of Hopf, was always to be applied
to geometry. That was the idea. It was not just something
abstract. One of the geometries was differential geometry,
manifolds. So, at an early stage I went through complex
manifolds, Kähler 2 manifolds, with my student Heinrich
Guggenheimer. We even created the name Kähler
manifold!
1
Samuel Eilenberg (1913–1998), Saunders Mac Lane (1909–
2005).
2
Erich Kähler (1906-2000).
3
Held 11–12 April 2007 at the ETH Zurich
Ah, that was your invention!
Indeed. The reason was that we used the operator of
Kähler on differential forms. Of course we used the book
by William Hodge (1903–1975). As I explained, the topology
of the classifying space of a group depends only
on the group. So at the same time, we had to develop the
formalism to work directly with the homology of groups,
and then the homology of algebras because groups lead
to group algebras; so we went to algebras. Then I went
on to dualize every map, considering a map in the other
direction as well. From this point of view, one obtains
new theorems. Pursuing this direction further on, I got
interested in groups by themselves: in applying geometry
to groups, topology to groups. And this then led to Poincaré
duality, Poincaré duality groups and duality groups.
It is a much more general setting that I developed in collaboration
with Robert Bieri. Together with many other
people and after a long development I could prove that
a Poincaré duality group of cohomological dimension 2
is the group of a Riemann surface. That was actually a
conjecture of Jean-Pierre Serre. “You have to prove it!”
he had always insisted.
In my earlier papers in topology, I had used cell complexes,
chains (which are linear combinations of cells)
and harmonic chains (which are cycles and cocycles at the
same time). There seemed to be something hidden, which
is typical for operator analysis. It was Jean-Pierre Serre
who always insisted: “There must be something much
deeper!” I did not know what it was for a long time.
Finally L 2 –theory came up, with idealized chains. Now
you can have harmonic chains inside that space of L 2 -
chains. And then you get so many things from earlier
considerations that guided me through L 2 -theory: topology
with Hilbert spaces instead of vector spaces, operators
– these were the things I worked on, until I more or
less stopped recently. Well, maybe not quite … I can still
read, for example I read what Wolfgang Lück has done.
You see, when Lück was very young, I got a paper from
him where he proved something I had just announced
also having proved.
There will be a meeting 3 next April for your 90th birthday.
Clearly you are still active. What is the driving
force that pushes you to continue doing mathematics?
The same force that was there at the beginning: because
I like it. I like it and I find it fascinating. I try to follow a
little bit of what the young people are doing, to understand
a little bit of what directions they go and how they
use the old stuff.
Coming back to your own contributions: is there one
single result that you view as your most important?
One single result - that is difficult! But if I would single
out something, it is probably what I did in the beginning.
It is so elementary today that it belongs to the first semester
of topology: homotopy groups, the exact sequence of
a fibration, calculating the homotopy groups for the orthogonal
groups and so on. There was nothing like that
before! And then I used the same homotopy methods to
prove that on a sphere of dimension 4k+1, you can only
have one tangent unit field, not two that are linearly independent.
It was the beginning of the whole theory of
EMS Newsletter December 2007 33

Interview
vector fields on spheres, which was developed by others
later on. It was very difficult in higher dimensions until
the famous papers by John Milnor and Michel Kervaire
(1927-2007). Milnor was not my student but I took him
from Princeton to Zurich, when he was a student. Kervaire
was my student but he wrote his thesis with Hopf.
At this stage, I should at least mention various geometric
and algebraic techniques introduced in algebraic
topology during my most active years, like spectral sequences,
cohomology operations, general homology theories
and so on. But in this conversation, I think we should
concentrate on comparatively elementary aspects.
Another direction was Eckmann-Hilton duality,
which went on for many years. There was even a section
in Mathematical Reviews under that name, with many
contributions.
Still another important area is Poincaré duality for
groups, invented by Robert Bieri and myself. They behave
like manifolds: homology, cohomology, you see, in
complementary dimensions, but with another dualizing
module. Many groups that are interesting in algebraic
geometry, group theory or other areas are such duality
groups. I had a draft of a paper about these topics with
me in Princeton and Jean-Pierre Serre was there at the
same time. He could not come to my lecture but the day
after he wrote to me that he wanted to publish it in the
Inventiones! It was not ready to be published yet – two
months were still necessary.
I don’t know what is very important, what is less important.
What you do is always interesting; it is more
difficult to judge importance! And then, I had so many
students! I gave many ideas and interests to my PhD students
and they then published work that I could not have
thought about myself.
Students
You mentioned your many PhD students; indeed according
to the Mathematics Genealogy Project, you
had 60 PhD students and more than 600 descendants.
How did you manage?
That is a good question! I don’t know! The first of the students
was an assistant who wanted to write a PhD with
me. I told him to write down, in one or two weeks, what he
really wanted to do, an abstract. I told him to read a little
bit and after a long time, maybe half a year or even more,
he should come and tell me what he really wanted to do.
And once he had his topic, we would see each other, for
one or two hours, and discuss things in detail, and start to
write down first results. Then I had the next student, and
the next, and the next … more and more.
Is there a particular reason why you attracted so many
students?
I don’t know! I mean, it’s probably because they liked
my style. In fact, I gave many lectures. At that time we
gave more lectures than today. To teach, you must make
it very clear in your mind what you want to lecture about,
how to present it and what to say first, and then you head
towards a result, a theorem.
My lecturing style is very old-fashioned, and probably
young people do not agree with me; that’s normal!
When lecturing, I always used blackboard and chalk, developing
the ideas gradually further and further, saying
exactly where I wanted to go. Sometimes I had to lecture
with overhead projectors. But then I wrote maybe four
or five lines and I would cover all the rest, except the one
line that I would have written on the blackboard. So it’s
really old-fashioned but I know there are many mathematicians
who still organise their lectures in that way.
My students seemed to like it because they followed my
courses. I did not allow any script, mimeographed notes
or anything. I said: “You have to think here and I go with
you step by step.” When the course is finished, you must
get the book and read it, and you will find other similar
things.
Today many students just use the mimeographed
notes. They have their colour pens and they underline
this and that; I don’t think it’s the same. But it works as
well! Today also, my colleagues find good students and
they have good PhDs. It’s just different!
Collaboration
Among your many collaborators, Peter Hilton clearly
plays a privileged role. Can you say some words about
the way you did your joint work?
I met Peter Hilton when he was a graduate student
with Henry Whitehead (1904–1960) at Oxford. I went in
1947 from the IAS to Oxford to meet Whitehead. Peter
was very shy then and he asked me whether he could
contact me at Zurich later on. I agreed and so he did.
In 1955 he came to Zurich and he stayed here for the
whole year. I could guide him a little bit and explain
many things to him about homological algebra, and then
he got more and more into that idea of dualizing lots
of our mathematics. And this became Eckmann-Hilton
duality, which was quite well followed for a while: in
geometry, topology and algebra. When he left Zurich,
we continued of course by correspondence. Sometimes
I went to England, or he came to Zurich and that was
alright. After a long time, he changed direction and I
always had the wish to do more concrete mathematics,
more geometry, more group theory; so we took different
routes. Our minds were a bit different and that was
alright: we remained very good friends but we did not
collaborate after that.
In your long career, you met quite a number of famous
mathematicians. Is there anybody whom you
would like to mention in terms of influence, or friendship?
I already mentioned Peter Hilton and Robert Bieri.
Then there is Guido Mislin; we have joint papers on
Chern classes of group representations. This work again
combines topology with group theory and with number
theory because the Chern class gives really interesting
limitations related to Bernoulli numbers and so on; it’s
an interesting topic!
34 EMS Newsletter December 2007

Interview
These were collaborators with whom I wrote joint
papers. Georges de Rham was very important for me in
Lausanne, and also afterwards; I went to see him from
time to time. But then I got of course a lot of very lucky
influence very early on from Henri Cartan 4 , who is already
more than one hundred years old, and later on
from Jean-Pierre Serre. Actually Jean-Pierre Serre is
younger than I am; he has followed my first papers very
carefully and that was really an interesting advantage. I
would always have wonderful contact with him; he asked
important questions. Unfortunately, I could not follow
him any longer when he went into number theory.
Forschungsinstitut für Mathematik
I would like to ask you about FIM, the Forschungsinstitut
für Mathematik, which you founded at the ETH,
and of which you were the first director for 20 years.
What was the prime idea for creating this institute?
How did it develop? Are you satisfied with its present
activities?
Indeed, I founded it because I thought it was necessary
to have an organization to welcome visitors and to do
everything so that they can work here together with faculty
members. The institute was not to have permanent
members, except for the director who had to be one of
the faculty members.
Something like that did not exist previously. The Institute
for Advanced Study was separated from Princeton
University and was not linked to it. It was essential for me
that our institute was to be linked with the department
here so that every member of the department could invite
visitors to work with or to learn from. And the institute
should care for these visitors in every respect. That was
an idea that people found strange at the time and many
did not agree. I went with this idea to the ETH president
Hans Pallmann. I argued that we needed such an institute
because otherwise our professors do not have enough
interaction with the world outside. He said: “We do not
have the money, but you get it! Just start right away!”
Soon afterwards, I could invite K. Chandrasekharan and
Lipman Bers (1914–1993). Many others followed.
I remained the director for twenty years. At the end
we had a huge number of visitors. My successor was Jürgen
Moser (1928–1999). He had a different style but he
worked towards the same objectives. He was followed
by Alain-Sol Sznitman and the present director is Marc
Burger. I think it will continue in the same spirit, although
many new features have been added, for example
the Nachdiplomvorlesungen (post-diploma lectures):
we invite people to the institute to give very high level
graduate courses 5 .
We have two or three such courses every semester.
Nowadays, they organise workshops as well. All this
4
Born in 1904.
5
In 1999, one of the interviewers, A.V., gave such a Nachdiplomvorlesung
on the Baum-Connes conjecture. This led to a very
stimulating interaction with Profs Eckmann and Mislin.
changed the size of the institute, of course; it has grown.
But the institute still takes care of apartments for visitors
and for their offices within the ETH.
The director Marc Burger has an excellent knowledge
of mathematics and of mathematicians all over the
world, so he attracts good visitors. Moreover, with all our
later appointments of high level mathematicians to the
department, people expressed interest in the institute
during negotiations: “Can I invite people to work with
my PhD-students?” It is quite important and I am very
pleased.
Nowadays, almost every university has such an institute
but at that time, in 1964, there was not a single one
– nowhere!
Publishing mathematics
Can we talk about your involvement in the publishing
of mathematics? For many years, you have been an editor
of the series Grundlehren der Mathematischen Wissenschaften
and also of Lecture Notes in Mathematics.
I was asked to join the managing board of the Grundlehren
because they needed people. Wolfgang Schmidt who
was there wanted to retire and van der Waerden said that
he no longer wanted to do that much.
At what time did you join Grundlehren?
That was in 1966. Every volume was refereed before being
accepted and this was heavy work.
Konrad Springer, 4 th generation of Springer, was a biologist
who studied in Zurich and I talked with him about the
publishing of lecture notes. The institute published lots of
lecture notes. Who could be a commercial publisher of such
notes? Springer-Verlag, of course! I talked to him and said:
“That is something I have wished for a long time, so if you
help me…” He tried to convince his father and they finally
liked the idea. They made photocopies of the typescript
and published it! You could send the typescript directly
to Springer who provided the copies. It was immediately
a great success. Since I could not take both series myself,
I asked Albrecht Dold to take over the Lecture Notes; the
first volume is under him. But he did not want to do the
work alone. He argued that I knew all the Springer people
and convinced me that we should both be editors.
Over the years it became easier, with computers and
the internet; it certainly takes less time. Now they receive
a computer-processed manuscript, only one or two referees
need to accept it and it runs very quickly. The contact
with Springer was always very interesting; we had
long discussions over many years.
IMU
You were also very much involved in national and international
mathematical societies. You have been the
President of the Swiss Mathematical Society for a two
year period…
That was almost compulsory; I had to do that…
EMS Newsletter December 2007 35

A MERICAN
M ATHEMATICAL SOCIETY
Releases from the AMS
Roots to Research
A Vertical Development of Mathematical
Problems
Stochastic Processes
S. R. S. Varadhan, Courant Institute of
Mathematical Sciences, New York, NY
Judith D. Sally, Northwestern University,
Evanston, IL, and Paul J. Sally, Jr.,
University of Chicago, IL
A guide to five of the most studied mathematical
problems, from their elementary origins to results
in current research
2008; 340 pages; Hardcover; ISBN: 978-0-8218-4403-8; List
US$49; AMS members US$39; Order code MBK/48
An introduction to stochastic processes from a
leading contributor to probability theory
Titles in this series are co-published with the Courant
Institute of Mathematical Sciences at New York University.
Courant Lecture Notes, Volume 16; 2007; 126 pages;
Softcover; ISBN: 978-0-8218-4085-6; List US$29; AMS
members US$23; Order code CLN/16
A View from the Top
Analysis, Combinatorics and Number
Theory
Alex Iosevich, University of Missouri,
Columbia, MO
The Ricci Flow: Techniques and
Applications
Part II: Analytic Aspects
Bennett Chow, Sun-Chin Chu, David
Glickenstein, Christine Guenther, James
Isenberg, Tom Ivey, Dan Knopf, Peng Lu,
Feng Luo, and Lei Ni
A demonstration of mathematics’ diversity and the
interconnectedness of its many subject matters
A detailed but accessible account of the
analytic techniques and results of Ricci flow
Student Mathematical Library, Volume 39; 2007; 136
pages; Softcover; ISBN: 978-0-8218-4397-0; List US$29; AMS
members US$23; Order code STML/39
Mathematical Surveys and Monographs, Volume 144;
2008; approximately 470 pages; Hardcover; ISBN: 978-0-
8218-4429-8; List US$109; AMS members US$87; Order
code SURV/144
COURANT
S. R. S. VARADHAN
Stochastic
Processes
16
LECTURE
NOTES
Ricci Flow and the Poincaré
Conjecture
John Morgan, Columbia University, New
York, NY, and Gang Tian, Princeton
University, NJ, and Peking University,
Beijing, China
Full details of a complete proof of the important
Poincaré Conjecture, using geometric arguments
presented by Grigory Perelman
Titles in this series are co-published with the Clay
Mathematics Institute (Cambridge, MA).
Harmonic Analysis on Commutative
Spaces
Joseph A. Wolf, University of California,
Berkeley, CA
An efficient introduction to commutative space
theory, emphasizing the analytic approach to
the theory
Mathematical Surveys and Monographs, Volume 142;
2007; 387 pages; Hardcover; ISBN: 978-0-8218-4289-8; List
US$99; AMS members US$79; Order code SURV/142
American Mathematical Society
Courant Institute of Mathematical Sciences
Clay Mathematics Monographs, Volume 3; 2007; 521
pages; Hardcover; ISBN: 978-0-8218-4328-4; List US$69; AMS
members US$55; Order code CMIM/3
1-800-321-4AMS (4267), in the U. S. and Canada, or 1-401-455-4000 (worldwide); fax:1-401-455-4046; email: cust-serv@ams.org.
American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA
For many more publications of interest,
visit the AMS Bookstore
www.ams.org/bookstore

Interview
…and secretary of the IMU…
Well, that was not compulsory. Heinz Hopf was IMU’s
president and he asked me to become the secretary of
the international union. I said: “Yes, if I can have a helping
secretary, because I do not have a secretary here!”
This is how I got a secretary to do the typing and mailing
for me. It was a very interesting period, 1956–1961.
What were the important issues at the time, during the
cold war?
Two important goals were achieved:
Many countries (some of them very large) that had not
adhered to the IMU became members. One can imagine
that many difficulties had to be overcome, difficulties of
personal, political and financial character. This was heavy
but gratifying work for the secretary.
The International Congress of Mathematicians had
to become a task of the IMU. The last congress organised
solely by a single country was the Congress in Edinburgh in
1958, organised by the UK. With the increasing number of
research areas and of participants, this became too heavy
a burden for a national mathematical society. The local organisation
is of course still taken care of by the organising
country but the scientific plans are made by the union.
Private Interests
What are your other main private interests – apart from
mathematics?
Through my entire mathematical life I was always able to
find time for other activities (sometimes combined with
mathematical work): I spent interesting periods with my
wife and my big family, on weekends, during vacations,
with music and art, and with school and student problems.
Love and happiness are important inside and outside
mathematics.
Thank you very much for this interesting conversation!
A Survey of ICMI Activities
Maria G. (Mariolina) Bartolini Bussi (Modena, Italy)
Maria G. (Mariolina) Bartolini Bussi is
a member of the editorial board of this
newsletter and serves as a member of the
executive committee of the International
Commission on Mathematical Instruction
from 01 January 2007 until 31 December 2009. In
this column, she will periodically provide news from the
ICMI.
The information below is taken from the official website
of the eleventh International Congress on Mathematical
Education (ICME11), which is to be held in Monterrey,
Mexico, 6–13 July 2008. The interested reader is
welcome to visit the website (http://icme11.org/), where
the second announcement will appear over the next few
weeks. Below is a summary of the session types given.
Most activities (topic study groups, discussion groups,
workshops, sharing experiences groups, a poster exhibition
and round tables) welcome contributors. Each activity
will have its own deadline, but not later than 20
January 2008.
The organizers expect to gather between 3000 and 4000
professionals from 100 countries in the mathematics education
area, including researchers, educators and teachers.
The International Congress on Mathematical Education
(ICME) aims to:
- Show what is happening in mathematics education
worldwide, in terms of research as well as teaching
practices.
- Inform about the problems of mathematics education
around the world.
- Learn and benefit from recent advances in mathematics
as a discipline.
ICME consists of several different session types.
Plenary Activities
Lectures or panels on themes of current actuality and
relevance to the practice of the international community
of mathematics educators.
National Presentations
It is customary to select a small number of countries so
that the international mathematics community may gain
a closer knowledge on the state and trends of mathematics
education in those countries.
National representatives of those countries are asked
to make the presentations.
EMS Newsletter December 2007 37

Mathematics Education
Survey Teams
ICME 11 survey teams, first created in ICME 10, are
groups entrusted to carry out a survey of the latest developments
regarding a certain theme or issue of mathematics
education. Emphasis is placed on pinpointing
new knowledge, new perspectives and emerging challenges.
The teams’ work will be presented in a lecture at
the congress. Survey teams ensure we are made aware
of developments in the field addressed since the time
of the previous ICME, thus giving continuity to the
ICME.
- ST 1: Recruitment, entrance and retention of students
to university mathematics studies in different countries.
- ST 2: Challenges to mathematics education research
faced by developing countries.
- ST 3: The impact of research findings in mathematics
education on students’ learning of mathematics.
- ST 4: Representations of mathematical concepts, objects
and processes in mathematics teaching and learning.
- ST 5: Mathematics education in multicultural and multilingual
environments.
- ST 6: Societal challenges to mathematics education in
different countries.
- ST 7: The notion and role of theory in mathematics
education research.
Regular Lectures
These lectures will be presented by experts invited by the
International Program Committee (IPC).
Topic Study Groups (TSGs)
The purpose of a TSG is to gather participants interested
in a certain topic of mathematics education. The organising
team of each TSG will review, select and organise
contributions, some by invitation and some submitted
by interested participants, that account for advances,
new trends and important work done in the last few
years on the topic the TSG addresses. The contribution
selected will be made available in one or more of the
following forms: as a download from the web page of
the TSG on the ICME11 web site; as a printed handout
prior to a TSG session during the congress; or by
oral presentation during any of the four TSG sessions.
There will be some discussion during the sessions but
emphasis is on presentation (in contrast to discussion
groups).
- TSG 1: New developments and trends in mathematics
education at preschool level.
- TSG 2: New developments and trends in mathematics
education at primary level.
- TSG 3: New developments and trends in mathematics
education at lower secondary level.
- TSG 4: New developments and trends in mathematics
education at upper secondary level.
- TSG 5: New developments and trends in mathematics
education at tertiary level.
- TSG 6: Activities and programs for gifted students.
- TSG 7: Activities and programs for students with special
needs.
- TSG 8: Adult mathematics education.
- TSG 9: Mathematics education in and for work.
- TSG 10: Research and development in the teaching
and learning of number systems and arithmetic.
- TSG 11: Research and development in the teaching
and learning of algebra.
- TSG 12: Research and development in the teaching
and learning of geometry.
- TSG 13: Research and development in the teaching
and learning of probability.
- TSG 14: Research and development in the teaching
and learning of statistics.
- TSG 15: Research and development in the teaching
and learning of discrete mathematics.
- TSG 16: Research and development in the teaching
and learning of calculus.
- TSG 17: Research and development in the teaching
and learning of advanced mathematical topics.
- TSG 18: Reasoning, proof and proving in mathematics
education.
- TSG 19: Research and development in problem solving
in mathematics education.
- TSG 20: Visualization in the teaching and learning of
mathematics.
- TSG 21: Mathematical applications and modeling in
the teaching and learning of mathematics.
- TSG 22: New technologies in the teaching and learning
of mathematics.
- TSG 23: The role of history of mathematics in mathematics
education.
- TSG 24: Research on classroom practice.
- TSG 25: The role of mathematics in the overall curriculum.
- TSG 26: Learning and cognition in mathematics - students’
formation of mathematical conceptions, notions,
strategies and beliefs.
- TSG 27: Mathematical knowledge for teaching.
- TSG 28: In-service education, professional life and development
of mathematics teachers.
- TSG 29: The pre-service mathematical education of
teachers.
- TSG 30: Students’ motivation and attitudes towards
mathematics and its teaching.
- TSG 31: Language and communication in mathematics
education.
- TSG 32: Gender and mathematics education.
- TSG 33: Mathematics education in a multilingual and
multicultural environment.
- TSG 34: Research and development in task design and
analysis.
- TSG 35: Research on mathematics curriculum development.
- TSG 36: Research and development in assessment and
testing in mathematics education.
- TSG 37: New trends in mathematics education research.
- TSG 38: The history of the teaching and learning of
mathematics.
38 EMS Newsletter December 2007

Mathematics Education
Discussion Groups (DGs)
DGs are meant to gather congress participants who wish
to actively discuss, in a genuinely interactive way, certain
challenging or controversial issues and dilemmas of a substantial,
non-rhetorical nature pertaining to the theme of
the DG. During the year from now up to the congress, the
discussion group will post contributions on their page on
the ICME11 web site that define, limit, and/or present
basic premises, viewpoints, theoretical considerations, research
findings and facts that should be accounted for if
a fruitful discussion is to be attained.
- DG 1: Curriculum reform – movements, processes and
policies.
- DG 2: The relationship between research and practice
in mathematics education.
- DG 3: Mathematics education – for what and why?
- DG 4: Re-conceptualizing the mathematics curriculum.
- DG 5: The role of philosophy in mathematics education.
- DG 6: The nature and roles of international cooperation
in mathematics education.
- DG 7: Dilemmas and controversies in the education of
mathematics teachers.
- DG 8: The role of mathematics in access to tertiary
education.
- DG 9: Promoting creativity for all students in mathematics
education.
- DG 10: Public perceptions and understanding of mathematics
and mathematics education.
- DG 11: Quality and relevance in mathematics education
research.
- DG 12: Rethinking doctoral programs in mathematics
education.
- DG 13: Challenges posed by different perspectives, positions
and approaches in mathematics education research.
- DG 14: International comparisons in mathematics education.
- DG 15: The shaping of mathematics education through
assessment and testing.
- DG 16: The evaluation of mathematics teachers and
curricula within educational systems.
- DG 17: The changing nature and roles of mathematics
textbooks - form, use, access.
- DG 18: The role of ethno-mathematics in mathematics
education.
- DG 19: The role of mathematical competitions and
other challenging contexts in the teaching and learning
of mathematics.
- DG 20: Current problems and challenges in primary
mathematics education.
- DG 21: Current problems and challenges in lower secondary
mathematics education.
- DG 22: Current problems and challenges in upper secondary
mathematics education.
- DG 23: Current problems and challenges in non-university
tertiary mathematics education.
- DG 24: Current problems and challenges in university
mathematics education.
- DG 25: Current problems and challenges in distance
teaching and learning.
- DG 26: Current problems and challenges in the conditions
and practice of mathematics teachers.
- DG 27: How is technology challenging us to re-think
the fundamentals of mathematics education?
- DG 28: The role of professional associations in mathematics
education - locally, regionally and globally.
Workshops
Workshops will provide hands-on experience to delegates
wishing to learn new skills. These
workshops are created via proposals to the IPC.
Sharing Experiences Groups (SEGs)
SEGs are small and intimate groups designed to exchange
and discuss experiences pertaining to research and/or
teaching. SEGs are formed via proposals to the IPC.
Poster Exhibition and Round Tables
Participants are invited to submit proposals for the display
and presentation of posters.
Round tables will address groups of posters developed
on the same theme.
Latin America: Perspectives on development
through collaboration
Concurrent with other ICME11 activities, we will organize
meetings that will address the issue of Latin American
development and collaboration. In spite of their differences,
Latin American countries share cultural roots, ethnic
diversity and a sense of identity. We wish to provide
a forum where participants will explore the possibilities,
advantages and perils of development through collaboration,
not only within Latin America but also with other
regions.
Mariolina Bartolini Bussi [bartolini@
unimo.it]
is the Newsletter Editor within Mathematics
Education. A short biography can be
found in issue 55, page 4.
40 EMS Newsletter December 2007

Cyprus Mathematical Society
Societies
1983–2008, 25 years of contribution and development to mathematical
science and education in Cyprus and Europe
Gregory Makrides (Nicosia, Cyprus)
Attempts to found the Cyprus Mathematical Society
started long before 1983. However, in 1983 a small group
of mathematicians took the initiative and at a meeting
of the Higher Technical Institute of Cyprus decided to
establish the Cyprus Mathematical Society. On 27 October
1983, the Cyprus Mathematical Society (CMS) was
born. In the founding statement it was reported that “the
objectives of CMS are the promotion and upgrading of
mathematical science and education”. The CMS decided
to use as its emblem the Cypriot Philosopher Zenon of
Kition (333 BC–264 BC) from Citium of Cyprus.
The CMS began with limited funds. Characteristically,
for initial international obligations we could not send a
complete team because of lack of resources. Today, the
CMS owns two offices, employs a secretary, offers scholarships
and meets without particular difficulty its economic
obligations, with an ultimate goal of establishing a
large Mathematics Centre of Excellence and a library.
The first president of the CMS was the inspector of
mathematics Mr Panagiotis Michael. The second president
Mr Glafkos Antoniades, who was also an inspector
of mathematics, was elected in 1990, and the current
president Dr Gregory Makrides, who is the Director of
Research and International Relations at the University
of Cyprus, was elected in 1998.
Main actvities of the Cyprus Mathematical Society
The activities of the Cyprus Mathematical Society have
progressively increased over the years both in quantity
and quality. In the presentation below we will focus on
the end product, which is the comprehensive situation
over the last decade.
1. Local competition
1.1 National competition of mathematics
The Cyprus Mathematical Society organises many different
regional and national mathematics competitions.
There is one process of competitions that progressively
converges to the selection of the national teams of Cyprus
that participate in different International Olympiads.
The sequence starts in November each year and
continues until May, at which time the national teams for
different International Olympiads are selected.
1.2 Cyprus Mathematical Olympiad
On 08 January 2000, because the year 2000 was announced
by UNESCO as the “year of mathematics”, the
CMS organised a new type of competition, called the
“Cyprus Mathematical Olympiad (CMO)”, using multiple
choice tests and which, for the first time in Cyprus
history, was open to students from 4 th Grade up to 12 th
Grade (ages 9 to 18). The attendance and participation
of students exceeded expectations. Over 4000 students
participated in this contest and because of this unexpected
interest it was decided to make it an annual event.
Since then the number of contestants has nearly reached
12000 students, which is about 15% of the national student
body. The award ceremony for this Olympiad attracts
more than 3000 people. The CMO as an activity
appeared in the European Commission’s report in 2001,
which reported five activities that took place in countries
in Europe that contributed to the advancement of
mathematics.
1.3 Mathimatiki Skytalodromia (Mathematical
Relay Race)
In 2007 the Cyprus Mathematical Society decided to celebrate
the “Day of Learning” with the introduction of a
new competition, which would include some gaming activity.
This was a competition between schools and groups
of students with the aim of promoting cooperative learning
and team competition. The competition was trialled
in 2007 in one district of Cyprus between 13 gymnasiums
(grades 7 to 9). The national competition, a “Mathematics
Relay Race” will run for the first time on 1 February
2008.
2. International competitions
2.1 Balkan Mathematical Olympiad (BMO)
In recent years, Cyprus has become active in hosting many
international events relating to mathematics, with some
of them initiated by the CMS. The CMS has organised
four Balkan Mathematical Olympiads: the 5 th in 1988, the
10 th in 1993, the 15 th in 1998 and the 23 rd in 2006.
2.2 Junior Balkan Mathematical Olympiads for
students under the age of 15 years (JBMO)
Since the first JBMO in 1997, the CMS has participated
every year. It organised the 5th JBMO in 2001 in Cyprus,
which was a great success. The 2007 JBMO was held in
Shumen, Bulgaria. It is expected that Cyprus will organise
the 2009 JBMO.
2.3 International Mathematical Olympiad (IMO)
From the first year of operation of the CMS in 1984, Cyprus
was invited to the IMO and participated with a full
team. Since then Cyprus, via the CMS, has participated in
almost all Mathematical Olympiads. Our first participa-
EMS Newsletter December 2007 41

Societies
tion was at the 25 th IMO that was organised in Prague in
1984 and we have participated in all IMOs since including
the 48 th in 2007 in Hanoi, Vietnam. The CMS is making
plans to host a future IMO in Cyprus.
2.4 International Mathematical Olympiad – Primary
Since 2001, Cyprus has participated in a special Olympiad
for primary school students. In 2007 we participated
in an “International Youth Mathematics Contest” with a
section called EMIC: Elementary Mathematics International
Contest. This was hosted in Hong Kong in late July
2007.
2.5 SEEMOUS: Mathematical Olympiad for
University Students
Through its active membership in the Mathematical Society
of South Eastern Europe (MASSEE), for which the
CMS holds one of the vice-president positions, CMS proposed
the development of a mathematical Olympiad for
first and second year university students. The proposal
materialised as SEEMOUS (South Eastern European
Mathematical Olympiad for University Students) with
international participation. The first event SEEMOUS
2007 was hosted in Cyprus in March 2007. The results appear
on www.seemous.org.
3. Conferences
3.1 Cyprus national conference on mathematical
science and education
This conference was initiated by the CMS and in 2008
we will organise the 10 th national conference. This conference
has grown in size to 350 participants in 2007 and
includes sessions covering all levels of education.
3.2 Mediterranean conferences on mathematics
education
This is another series of conferences that was established
by the Cyprus Mathematical Society. The first
and second took place in Cyprus, in 1997 and 2000 respectively,
the third was hosted in Athens by the Hellenic
Math Society in 2003, the fourth was hosted by
the University of Palermo in Italy in 2005 and the fifth
was hosted by the University of the Aegean in Greece
in 2007. It is expected that the sixth will be hosted in
Bulgaria in 2009. The conferences have grown with the
participation of some 200 mathematicians from around
the world.
3.3 National student conference
The first Student Conference in Mathematics was organised
in February 2005 with the participation of about 90
students from grades 7 to 12 with 40 presentations given.
The event has become very popular and in 2007 there
were 350 students with 95 presentations given. Students
present mathematical projects and studies of all kinds.
The Cyprus Mathematical Society has decided to attempt
the organisation of a European Student Conference in
Mathematics in February 2009. The announcement of
this conference is expected to be circulated throughout
Europe in January 2008.
4. Seminars and training courses for adults
4.1 Seminars
The CMS organises European Grundtvig courses under
the title, “Preparation of Full Proposals and Management
of European Education Projects”. The courses appear
on the Europa course database ec.europa.eu/education/trainingdatabase.
The CMS was also an associate partner in the well
known MATHEU project, “Identification, Motivation
and Support of Mathematical Talents in European
Schools”. The project ran from 2003 to 2006 and now the
CMS helps in the organisation of the in-service training
courses for teachers. The website of the project is www.
matheu.eu and information on the courses can be found
at ec.europa.eu/education/trainingdatabase.
5. Publications
5.1 One of the most popular student periodicals has been
published by the Cyprus Mathematical Society since
1984. Its name is “Mathematiko Vima” and it is published
annually. During the last two years the periodical has
been divided into two volumes A and B by separating
its content so that volume A is of interest to all students
and volume B is more aimed at those who have a special
interest in mathematics.
5.2 An important scientific step was taken by the Cyprus
Mathematical Society when it decided to develop
a scientific journal on mathematics education called the
“Mediterranean Journal for Research Mathematics Education”.
The publication began in 2003 and two issues
are published per year. It is becoming well-known and is
gaining an international reputation.
42 EMS Newsletter December 2007

Societies
5.1 Other publications of CMS include:
- The proceedings of all the annual national conferences
in mathematics education.
- The proceedings of all the annual student conferences.
- The proceedings of the five Mediterranean conferences
on mathematics education.
- “Mathematics for Competitions”, published in English
in 2006.
6. Other Activities
6.1 Summer mathematics school
Since the summer of 1991, the CMS has been organising
a summer mathematics school with the aim of increasing
the interest of the student body in mathematics and to
show them the fun that can be had while studying mathematics.
The summer school accepts students from grades
7 up to 10 only. The summer school has grown from about
100 students in 1991 to 1200 in 2007 with applications of
more than 2500.
6.2 Prize to the best graduating university student
in mathematics
The CMS has established an annual award of 850 Euro
for the best graduating mathematics student from the
University of Cyprus.
6.3 Television education
One of the activities organised by the CMS in the past
was a series of television programmes on mathematics
offering lectures to help students prepare for the national
university entrance examinations. This programme was
sponsored by the national television station and it ran for
two years, 1995 and 1996.
6.4 National career day
The CMS participates every year in the career day, which
is organised by the Association of Career Guidance teachers
in Cyprus. We have our own booth and we distribute a
leaflet explaining what kinds of jobs a mathematician can
do and the importance of mathematics in all studies.
6.5 The CMS knows how to honour friends and
collaborators
The CMS rewards, on an annual basis, retiring mathematics
teachers. We also honour government officials, especially
from the Ministry of Education and Culture, who
for years have been cooperating in supporting the activities
of the society. We have also established the “Zenon
Award”, which is given to a Cypriot mathematician who
has made a special contribution to the advancement of
mathematics. This award has only been given once so
far, to the Cypriot professor Dimitris Christodoulou at
Princeton University, USA, for his contribution to mathematical
science. Other awards include the honorary
presidency offered to the first two presidents of the Cyprus
Mathematical Society.
6.6 Member of the European Mathematical Society
The Cyprus Mathematical Society is a member of the
European Mathematical Society (EMS). Its president
participates in the Education Committee of the EMS.
Relations between the two societies are expected to grow
as EMS has accepted an invitation to collaborate on the
new European Student Conference on Mathematics as
well as on summer mathematics schools.
It should be noted that the Cyprus Mathematical Society
is a non-profit organisation, and has regular members,
special members and reciprocity members. It is currently
managed by a 15 person management council and
its current mixed membership is close to 800.
Dr Gregory Makrides [makrides.g@ucy.
ac.cy] has been the president of the Cyprus
Mathematical Society since 1998.
The address of the Cyprus Mathematical
Society is 36 Stasinou street, Office 102, Strovolos 2003,
Nicosia, Cyprus. Tel: +357-22378101. Fax: +357-22379122.
www.cms.org.cy, cms@cms.org.cy
EMS Newsletter December 2007 43

Personal column
Personal column
Please send information on mathematical awards and
deaths to the editor.
Awards
Claire Voisin (CNRS Jussieu, Paris) has been awarded the 2007
Ruth Lyttle Satter prize of the American Mathematical Society
“for her deep contributions to algebraic geometry”.
The Fermat Prize 2007 for Mathematics Research has been
awarded to Chandrashekhar Khare (Univ. of Utah) “for his
proof in collaboration with Jean-Pierre Wintenberger of Serre’s
modularity conjecture in number theory”.
The 2006 John von Neumann Theory Prize, the highest prize
given in the field of operations research and management science,
has been awarded to Martin Grötschel (Technical University
of Berlin), László Lovász (Eötvös Loránd University,
Budapest) and Alexander Schrijver (University of Amsterdam
and the CWI) “for their fundamental ground-breaking work in
combinatorial optimization”.
Hô Hai Phùng (University of Duisburg-Essen, Germany) has
been awarded the von Kaven Prize in Mathematics “in recognition
of his outstanding work on quantum groups”.
The 2007 Spring Prize of the Mathematical Society of Japan
(MSJ) has been awarded to Kenji Nakanishi (Kyoto University)
for his distinguished and fundamental contributions to the
study of nonlinear dispersive equations.
The 2007 Seki-Takakazu Prize has been awarded to the Institut
des Hautes Études Scientifiques (IHES, Bures-sur-Yvette,
France) for its outstanding contribution to establishing strong
relationships between mathematicians in Japan and France by
offering invaluable research-exchange opportunities for the
development of mathematics since 1958.
The 2007 MSJ Algebra Prize was awarded to Eiichi Bannai (Kyushu
University) for his contribution to the study of algebraic
combinatorics and to Kouta Yoshioka of Kobe University for his
contribution to the theory of moduli spaces of vector bundles.
Remco van der Hofstad (Eindhoven University of Technology,
Netherlands) has been awarded the 2007 Rollo Davidson Prize.
Van der Hofstad was honoured for his work in probability and
statistical mechanics.
Mikołaj Bojańczyk (Warsaw) was awarded the Kuratowski
Prize.
Łukasz Kosiński (Kraków) was awarded the first Marcinkiewicz
Prizes of the Polish Mathematical Society for students’
research papers.
Steven Rudich (Carnegie Mellon University) and Alexander
A. Razborov (Steklov Mathematical Institute, Moscow) were
named recipients of the Gödel Prize of the Association for
Computing Machinery (ACM). They were recognized for their
work on the P versus NP problem.
Bryan Birch (University of Oxford) has been awarded the De
Morgan Medal of the London Mathematical Society (LMS) in
recognition of his influential contributions to modern number
theory.
Béla Bollobás (University of Cambridge) has been awarded
the Senior Whitehead Prize from the LMS for his fundamental
contributions to almost every aspect of combinatorics.
Michael Green (University of Cambridge) received the Naylor
Prize and Lectureship in Applied Mathematics in recognition
of his founding work in superstring theory.
Four Whitehead Prizes were awarded to: Nikolay Nikolov (University
of Oxford and Imperial College, London; group theory),
Oliver Riordan (University of Cambridge; graph theory and
combinatorics), Ivan Smith (University of Cambridge; symplectic
topology) and Catharina Stroppel (University of Glasgow;
representation theory).
Paul Fearnhead (Lancaster University, UK) has been awarded
the 2007 Adams Prize from the University of Cambridge for
major contributions to several areas of computational statistics
and population genetics.
Ian Stewart (University of Warwick, UK) has been awarded the
2006 Premio Peano from the Associazione Subalpina Mathesis
in Turin, Italy, for the Italian translation of his book Letters to
a Young Mathematician.
Deaths
We regret to announce the deaths of:
Graham R. Allan (UK, 9.8.2007)
Jakow Baris (Belarus, 26.7.2007)
Joachim Bergmann (Germany, 14.3.2007)
Harry Burkill (UK, 9.4.2007)
Fokko du Cloux (France, 10.11.2006)
Paul Joseph Cohen (UK, 23.3.2007)
Carl Geiger (Germany, 15.6.2007)
Gisbert Hasenjäger (Germany, 2.9.2006)
Anthony Horsley (UK, 26.5.2006)
Ahmet Hayri Körezlioǧlu (Turkey, 26.6.2007)
Paulette Libermann (France, 10.7.2007)
Wladyslaw E. Lyantse (Ukraine, 29.3.2007)
Dietrich Morgenstern (Germany, 27.6.2007)
Gert H. Müller (Germany, 9.9.2006)
Wolfgang Müller (Germany, 19.12.2006)
Morris Newman (UK, 4.1.2007)
Mircea Puta (Romania, 26.7.2007)
Gareth Roberts (UK, 6.2.2007)
Felice L. Ronga (Switzerland, 22.5.2007)
Johann Schröder (Germany, 3.1.2007)
Atle Selberg (Norway, 6.8.2007)
John Todd (UK, 21.6.2007)
Klaus Wohlfahrt (Germany, 3.7.2007)
44 EMS Newsletter December 2007

Book review
Xavier Gràcia (Barcelona, Spain)
Music and mathematics.
From Pythagoras to fractals.
Edited by John Fauvel,
Raymond Flood and Robin Wilson.
200 pages, £18 (Paperback edition).
Oxford University Press, Oxford, 2003 and 2006.
ISBN: 0-19-851187-6 and 978-0-19-929893-8.
Leibniz described music
as a secret exercise in
arithmetic of the soul unaware
of its act of counting.
The relationship between
music and mathematics is
much older, having been
in existence since at least
the time of Pythagoras.
Nevertheless, there is a
need to remind people
from both sides about this
relationship. That seems
to be the purpose of this
book, which aims to demonstrate and analyse “the continued
vitality and vigour of the traditions arising from
the ancient beliefs that music and mathematics are fundamentally
sister sciences”, as stated in the preface. The
book is a collection of articles so let us begin by describing
their contents, paying special attention to the mathematical
aspects.
The book begins with an introductory article, “Music
and mathematics: an overview”, by Susan Wollenberg. It
contains some notes describing how the relationship between
music and mathematics has been perceived and
discussed through history and more particularly since the
seventeenth century.
Part I: Music and mathematics through history, contains
two articles. The first one is “Tuning and temperament:
closing the spiral”, by Neil Bibby. It explains one
of the most basic facts about the mathematical structure
of musical scales. As is well-known, the interval between
two notes is given by the ratio of their frequencies. The
most basic ratio is 2:1, the octave. Two notes an octave
apart are heard as equivalent so to construct a scale
other intervals are needed. The next basic interval is the
perfect fifth, with frequency ratio 3:2 – a specially pleasing
interval. The Pythagorean scale is constructed by
adding fifths. One of the fundamental problems in music
theory has always been how many fifths one should add
and how those then can be modified conveniently. This
resulted in the division of the octave into 12 equal semitones
that has pervaded Western music since the 19th
century.
Book review
The other article is “Musical cosmology: Kepler and
his readers”, by Judith V. Field. Seemingly Johannes Kepler
believed that geometry and musical harmony could
explain the structure of the Universe. This idea, drawn
from the ancient tradition of the music of the spheres,
was discussed shortly after by Marin Mersenne and
Athanasius Kircher.
Part II: The mathematics of musical sound, contains
three articles. The first one, “The science of musical
sound”, by Charles Taylor, gives a qualitative account of
some properties of sound, its perception and its production
by musical instruments.
The second article in this part, “Faggot’s fretful fiasco”,
by Ian Stewart, describes a historical accident. In
the 18th century, a craftsman Daniel Strähle devised a
simple way to determine the position of the frets on a
guitar, which in mathematical terms amounts to approximating
the 12th root of 2. However, this was dismissed
by a mathematician Jacob Faggot due to a regrettable
mistake in his calculations. The article discusses Strähle’s
proposal in terms of fractional approximations and continued
fractions.
Finally, “Helmholtz: combinational tones and consonance”,
by David Fowler, describes two of the many
contributions of Hermann Helmholtz to the science of
sound. One is of a psychological nature: combinational
tones, that is tones that are sometimes heard due to the
nonlinearity of the ear. The other one is his explanation
of the consonance associated with frequency ratios of
small integers in terms that are very close to our present
theories.
Part III: Mathematical structure in music, also consists
of three articles. The first one, “The geometry of music”,
by Wilfrid Hodges, studies music from a geometric
perspective. The basic dimensions of time and pitch constitute
a 2-dimensional space, subject to transformations
like translations and rotations. A piece of music may
contain motifs, each motif being considered as a subset
of the musical space. Motifs are analysed according to
their symmetry groups. In the same way, there is a classification
of friezes. Many musical examples are given to
illustrate all these possibilities.
The second paper in this part is “Ringing the changes:
bells and mathematics”, by Dermot Roaf and Arthur
White. Here, permutation groups and graph theory are
applied to solve the old problem of change-ringing, that
is to ring a set of bells in all possible orders, without repetition.
“Composing with numbers: sets, rows and magic
squares”, by Jonathan Cross, describes various usages of
numbers by some 20th century composers, from Arnold
Schoenberg to Iannis Xenakis. The mathematical models
described here are new sources of musical material and,
as the author reminds us, the results may be as inspired
or as mechanical as in any other musical system.
In Part IV: The composer speaks, two articles can be
found. “Microtones and projective planes”, by Carlton
Gamer and Robin Wilson, deals with microtonal music,
based on n equal divisions of the octave. More particularly,
they are interested in the case where n = k 2 –k+1, which
EMS Newsletter December 2007 45

Conferences
is the number of points (and lines) in a finite projective
plane. Then there is a connection between the musical
inversion and the duality of projective geometry.
Finally, “Composing with fractals”, by Robert Sherlaw
Johnson, describes a computer program that generates
musical patterns based on fractals.
The essays contained in the book concern very different
subjects and have different mathematical density
and depth. In this sense, the book lacks some coherence
and elegance. Nevertheless, this allows the readers to
choose their own ways to enjoy some of the connections
between music and mathematics. Of course, many other
connections are almost absent: the theory of dissonance,
spectra of musical instruments, combinatorics of chords
and scales, rhythmic patterns, etc. But, in less than 200
pages, the book conveys the idea that both disciplines
have been closely related through history and that this
relationship is going to remain and even to increase.
Music searches for new ways of expression and mathematics
grows unceasingly. Therefore more and more
connections will show up between these disciplines. The
recent launch of a Journal of Mathematics and Music,
and the appearance of books like Dave Benson’s ‘Music:
a mathematical offering’ is a clear signal in this direction.
With respect to this, I would dare say that there lacks an
appropriate place in the Mathematics Subject Classification
for the many existing and forthcoming literature on
mathematics and music. Now that a revision of the classification
scheme is in progress, a three-digit section near
the end of the list could accommodate the many mathematical
faces of music.
Xavier Gràcia [xgracia@ma4.upc.edu]
has degrees in physics and mathematics
and obtained a PhD in physics in 1991
from the University of Barcelona. He has
been working since 1988 at the Technical
University of Catalonia in the Department
of Applied Mathematics IV. His research
interests are differential geometry
and its applications, especially to mathematical physics.
He is also an amateur musician and teaches a course on
Music and Mathematics.
Forthcoming conferences
compiled by Mădălina Păcurar (Cluj-Napoca, Romania)
Please e-mail announcements of European conferences, workshops
and mathematical meetings of interest to EMS members,
to one of the addresses mpacurar@econ.ubbcluj.ro or madalina_
pacurar@yahoo.com. Announcements should be written in a
style similar to those here, and sent as Microsoft Word files or as
text files (but not as TeX input files).
December 2007
3–7: Arithmetic geometry and rational varieties, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
7–8: The fifth conference on nonlinear analysis and applied
mathematics (CNAAM 2007), University Valahia, Targoviste,
Romania
Information: cmortici@valahia.ro
10–14: l-adic cohomology and number fields, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
17–21: Nonlinear equations and complex analysis, Ufa,
Russia
Information: bannoe07@matem.anrb.ru; http://matem.anrb.ru/
bannoe07
17–21: Meeting on mathematical statistics, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr; http://www.cirm.univ-mrs.fr
17–22: International Conference on Transformation
Groups, Moscow, Russia
Information: tg2007@mccme.ru; http://www.mccme.ru/tg2007/
27–30: The Second International Conference on Mathematics:
Trends and developments, Cairo, Egypt
Information: conf07@etms-web.org;
http://www.etms-web.org/conf07/
January 2008
6–13: 1st Odense Winter School on Geometry and Theoretical
Physics, University of Southern Denmark, Odense, Denmark
Information: swann@imada.sdu.dk;
http://www.imada.sdu.dk/~swann/Winter-2008/
7–11: Small group: D’Alembert’s Opuscules and mathematical
correspondences, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
14–18: Algebraic Geometry and Complex Geometry, CIRM
Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
46 EMS Newsletter December 2007

Conferences
21–24: Conference in Geometric Analysis and its Applications,
Bern, Switzerland
Information: geoan@math.unibe.ch; http://www.geoan.unibe.ch
21–25: International Conference on Uniform Distribution,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
28–31: VII International Conference “System Identification
and Control Problems” SICPRO ’08, Moscow, Russia
Information: sicpro@ipu.rssi.ru;
http://www.sicpro.org/sicpro08/code/e08_01.htm
28–February 1st: Holomorphic partial differential equations,
small divisors and summability, CIRM Luminy, Marseille,
France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
29–31: Workshop on integrability in the AdS/CFT correspondence,
Utrecht University, Utrecht, Netherlands
Information: l.k.hoevenaars@math.uu.nl;
http://www.math.uu.nl/adscft
February 2008
1–12: VIII Edition of the Russian Winter School, Kostroma,
Russia
Information: cdipietr@unisa.it, school08ru@diffiety.ac.ru;
http://school.diffiety.org/page3/page0/page63/page63.html
4–8: 2nd GNAMPA School on Harmonic Analysis and Evolution
Equations, Università di Parma, Parma, Italy
Information: alessandra.lunardi@unipr.it, mauceri@dima.unige.it;
http://www.unipr.it/arpa/dipmat/rivista/HAEE/HAEE.html
4–March 7: GREFI-MEFI, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
5–9: Second Winter School in Complex Analysis and Operator
Theory, Sevilla, Spain
Information: contreras@us.es; http://www.congreso.us.es/wscaot/
11–15: Fifth International Symposium on Foundations of
Information and Knowledge Systems, Pisa, Italy
Information: contact@foiks.org; http://2008.foiks.org/
12–16: Foundations of Lattice-Valued Mathematics with
Applications to Algebra and Topology, Linz, Austria
Information: ep.klement@jku.at;
http://www.flll.jku.at/research/linz2008
19–22: International Conference on Mathematics and
Continuum Mechanics, Porto, Portugal
Information: ferreira@fe.up.pt;
http://paginas.fe.up.pt/~cim2008/index.html
29–2 March: Joint Mathematical Weekend EMS-Danish
Mathematical Society, Copenhagen, Denmark
Information: www.math.ku.dk/english/research/conferences/
emsweekend/
March 2008
2–7: IX International Conference “Approximation and Optimization
in the Caribbean”, San Andres Island, Colombia
Information: appopt2008@univalle.edu.co;
http://matematicas.univalle.edu.co/~appopt2008/
3–5: International Technology, Education and Development
Conference (INTED2008), Valencia, Spain
Information: inted2008@iated.org; http://www.iated.org/inted2008
3–6: The Third International Conference On Mathematical
Sciences (ICM2008), UAE University – Al-Ain, United Arab
Emirates
Information: ICM2008@uaeu.ac.ae; http://icm.uaeu.ac.ae
4–7: 8th German Open Conference on Probability and
Statistics, Aachen, Germany
Information: gocps2008@stochastik.rwth-aachen.de;
http://gocps2008.rwth-aachen.de
5–8: The First Century of the International Commission
on Mathematical Instruction, Accademia dei Lincei, Rome,
Italy
Information: http://www.unige.ch/math/EnsMath/Rome2008/
6–April 4: Cours “Méthodes variationnelles et parcimonieuses
en traitement des signaux et des images”, Paris,
France
Information: gabriel.peyre@ceremade.dauphine.fr; http://www.
ceremade.dauphine.fr/~peyre/cours-ihp-2008/
9–12: LUMS 2nd International Conference on Mathematics
and its Applications in Information Technology 2008
(in collaboration with SMS, Lahore), Lahore, Pakistan
Information: http://www.lums.edu.pk/licm08
10–14: ALEA meeting, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
17–21: GL (2) , CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
19–21: The IAENG International Conference on Scientific
Computing 2008, Regal Kowloon Hotel, Kowloon, Hong Kong
Information: publication@iaeng.org;
http://www.iaeng.org/IMECS2008/ICSC2008.html
25–28: Taiwan-France joint conference on nonlinear partial
differential equations, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
EMS Newsletter December 2007 47

Does your research add up?
The Cambridge Journals mathematics list is growing all the time...with 5 new
journals in 2008, here are a few from our selection:
Pure mathematics
New for 2008
Journal of the Australian Mathematical
Society
Published for the Australian Mathematical Society
Bulletin of the Australian Mathematical
Society
Published for the Australian Mathematical Society
Journal of K-Theory
Published for the Independent Scholarly Online and
Print Publishing (ISOPP)
Glasgow Mathematical Journal
Published for the Glasgow Mathematical Journal Trust
Compositio Mathematica
Produced, marketed and distributed for the London
Mathematical Society for the Foundation Compositio
Mathematica
Proceedings of the Edinburgh
Mathematical Society
Published for the Edinburgh Mathematical Society
Mathematical Proceedings of the
Cambridge Philosophical Society
Published for the Cambridge Philosophical Society
Applied mathematics
New for 2008
The ANZIAM Journal
Published for the Australian Mathematical Society
European Journal of Applied
Mathematics
Review of Symbolic Logic
Published for the Association of Symbolic Logic
View our complete mathematics range by visiting
journals.cambridge.org/mathematics

Conferences
31–April 4: Mathematical Computer Science School for
Young Researchers, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
April 2008
6–9: Mathematical Education of Engineers, Mathematics Education
Centre, Loughborough University, Loughborough, UK
Information: mee2008@lboro.ac.uk; http://mee2008.lboro.ac.uk/
7–11: Graph decomposition, theory, logics and algorithms,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
14–18: Dynamical quantum groups and fusion categories,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
21–25: Multiscale methods for fluid and plasma turbulence:
Applications to magnetically confined plasmas in
fusion devices, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
28–May 2: Recent progress in operator and function theory,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
May 2008
5–9: Statistical models for images, CIRM Luminy, Marseille,
France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
12–16: New challenges in scheduling theory, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
19–21: ManyVal ’08 – Applications of Topological Dualities
to Measure Theory in Algebraic Many-Valued Logic,
Milan, Italy
Information: http://manyval.dsi.unimi.it/
19–23: Topological & Geometric Graph Theory, Paris,
France
Information: tggt2008@ehess.fr; http://tggt.cams.ehess.fr
19–23: Vorticity, rotation and symmetry – Stabilizing and
destabilizing fluid motion, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
26–30: Spring school in nonlinear partial differential equations,
Louvain-La-Neuve, Belgium
Information:
http://www.uclouvain.be/math-spring-school-pde-2008.html
26–30: The Fourth International Conference “Inverse Problems:
Modeling and Simulation”, Oludeniz-Fethiye, Turkey
Information: ahasanov@kou.edu.tr; http://www.ipms-conference.org/
26–30: Discrete Groups and Geometric Structures, with
Applications III, K.U. Leuven Campus Kortrijk, Belgium
Information: Paul.Igodt@kuleuven-kortrijk.be;
http://www.kuleuven-kortrijk.be/workshop
26–30: High dimensional probability, CIRM Luminy, Marseille,
France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
27–June 5: 5th Linear Algebra Workshop, Kranjska Gora,
Slovenia
Information: damjana.kokol@fmf.uni-lj.si; http://www.law05.si
28–31: History of Mathematics & Teaching of Mathematics,
Targu-Mures, Romania
Information: matkp@uni-miskolc.hu
29–31: Brownian motion and random walks in mathematics
and in physics, Institut de Recherche Mathématique
Avancée (Université Louis Pasteur), Strasbourg, France
Information: franchi@math.u-strasbg.fr, papadop@math.ustrasbg.fr;
http://www-irma.u-strasbg.fr/article545.html
June 2008
2–6: International Conference on Random Matrices
(ICRAM), Sousse, Tunisia
Information: abdelhamid.hassairi@fss.rnu.tn;
http://www.tunss.net/accueil.php?id=ICRAM
2–6: Thompson’s groups: new developments and interfaces,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
8–14: Mathematical Inequalities and Applications 2008,
Trogir - Split, Croatia
Information: mia2008@math.hr; http://mia2008.ele-math.com/
9–13: Geometric Applications of Microlocal Analysis,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
9–19: Advances in Set-Theoretic Topology: Conference in
Honour of Tsugunori Nogura on his 60th Birthday, Erice,
Sicily, Italy
Information: erice@dmitri.math.sci.ehime-u.ac.jp;
http://www.math.sci.ehime-u.ac.jp/erice/
16–20: Workshop on population dynamics and mathematical
biology, CIRM Luminy, Marseille, France
EMS Newsletter December 2007 49

Conferences
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
16–20: Homotopical Group Theory and Homological Algebraic
Geometry Workshop, Copenhagen, Denmark
Information: http://www.math.ku.dk/~jg/homotopical2008/
17–20: Structural Dynamical Systems: Computational Aspects,
Capitolo-Monopoli, Bari, Italy
Information: sds08@dm.uniba.it;
http://www.dm.uniba.it/~delbuono/sds2008.htm
17–22: International conference “Differential Equations
and Topology” dedicated to the Centennial Anniversary
of Lev Semenovich Pontryagin, Moscow, Russia
Information: pont2008@cs.msu.ru; http://pont2008.cs.msu.ru
22–28: Combinatorics 2008, Costermano (VR), Italy
Information: combinatorics@ing.unibs.it;
http://combinatorics.ing.unibs.it
23–27: Homotopical Group Theory and Topological Algebraic
Geometry, Max Planck Institute for Mathematics Bonn,
Germany
Information: admin@mpim-bonn.mpg.de;
http://www.ruhr-uni-bochum.de/topologie/conf08/
23–27: Hermitian symmetric spaces, Jordan algebras and
related problems, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
25–28: VII Iberoamerican Conference on Topology and its
Applications, Valencia, Spain
Information: cita@mat.upv.es; http://cita.webs.upv.es
30–July 3: Analysis, PDEs and Applications, Roma, Italy
Information: mazya08@mat.uniroma1.it;
http://www.mat.uniroma1.it/~mazya08/
30–July 4: Joint ICMI/IASE Study; Teaching Statistics in
School Mathematics. Challenges for Teaching and Teacher
Education, Monterrey, Mexico
Information: batanero@ugr.es;
http://www.ugr.es/~icmi/iase_study/
30–July 4: Geometry of complex manifolds, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
30–July 4: The European Consortium for Mathematics in
Industry (ECMI2008), University College, London, UK
Information: lucy.nye@ima.org.uk; http://www.ecmi2008.org/
July 2008
2–4: The 2008 International Conference of Applied and
Engineering Mathematics, London, UK
Information: wce@iaeng.org;
http://www.iaeng.org/WCE2008/ICAEM2008.html
3–8: 22nd International Conference on Operator Theory,
West University of Timisoara, Timisoara, Romania
Information: ot@theta.ro; http://www.imar.ro/~ot/
7–10: The Tenth International Conference on Integral
Methods in Science and Engineering (IMSE 2008), University
of Cantabria, Santander, Spain
Information: imse08@unican.es, meperez@unican.es;
http://www.imse08.unican.es/
7–11: VIII International Colloquium on Differential Geometry
(E. Vidal Abascal Centennial Congress), Santiago de
Compostela, Spain
Information: icdg2008@usc.es; http://xtsunxet.usc.es/icdg2008
7–11: Spectral and Scattering Theory for Quantum Magnetic
Systems, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
13 : Joint EWM/EMS Workshop, Amsterdam, Netherlands
Information: http://womenandmath.wordpress.com/joint-ewmems-worskhop-amsterdam-july-13th-2008/
14–18: Fifth European Congress of Mathematics (5ECM),
Amsterdam, Netherlands
Information: www.5ecm.nl
15-18: Mathematics of program construction, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
15–19: The 5th World Congress of the Bachelier Finance
Society, London, UK
Information: mark@chartfield.org; http://www.bfs2008.com
17–19: 7th International Conference on Retrial Queues
(7th WRQ), Athens, Greece
Information: aeconom@math.uoa.gr;
http://users.uoa.gr/~aeconom/7thWRQ_Initial.html
20–23: International Symposium on Symbolic and Algebraic
Computation, Hagenberg, Austria
Information: franz.winkler@risc.uni-linz.ac.at;
http://www.risc.uni-linz.ac.at/about/conferences/issac2008/
21–24: SIAM Conference on Nonlinear Waves and Coherent
Structures, Rome, Italy
Information: meetings@siam.org;
http://www.siam.org/meetings/nw08/
21–August 29: CEMRACS, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
50 EMS Newsletter December 2007

Conferences
24–26: Workshop on Current Trends and Challenges in
Model Selection and Related Areas, University of Vienna,
Vienna, Austria
Information: hannes.leeb@yale.edu;
http://www.univie.ac.at/workshop_modelselection/
August 2008
3–9: Junior Mathematical Congress, Jena, Germany
Information: info@jmc2008.org; http://www.jmc2008.org/
16–31: EMS-SMI Summer School: Mathematical and numerical
methods for the cardiovascular system, Cortona,
Italy
Information: dipartimento@matapp.unimib.it
19–22: Duality and Involutions in Representation Theory,
National University of Ireland, Maynooth, Ireland
Information: involutions@maths.nuim.ie;
http://www.maths.nuim.ie/conference/
September 2008
1–5: Representation of surface groups, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
2–5: X Spanish Meeting on Cryptology and Information
Security, Salamanca, Spain
Information: delrey@usal.es;
http://www.usal.es/xrecsi/english/main.htm
8–12: Chinese-French meeting in probability and analysis,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
8–19: EMS Summer School: Mathematical models in the
manufacturing of glass, polymers and textiles, Montecatini,
Italy
Information: cime@math.unifi.it;
http://web.math.unifi.it/users/cime//
15–19: Geometry and Integrability in Mathematical Physics
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
21–24: The 8th International FLINS Conference on Computational
Intelligence in Decision and Control (FLINS
2008), Madrid, Spain
Information: flins2008@mat.ucm.es;
http://www.mat.ucm.es/congresos/flins2008
29–October 3: Commutative algebra and its interactions
with algebraic geometry, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
29–October 8: EMS Summer School: Risk theory and related
topics,
Będlewo, Poland
Information: stettner@iman.gov.pl;
www.impan.gov.pl/EMSsummerSchool/
October 2008
6–10: Partial differential equations and differential Galois
theory, CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
13–17: Hecke algebras, groups and geometry, CIRM Luminy,
Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
20–24: Symbolic computation days, CIRM Luminy, Marseille,
France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
27–31: New trends for modeling laser-matter interaction,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
November 2008
3–7: Harmonic analysis, operator algebras and representations,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
5–7: Fractional Differentiation and its Applications, Ankara,
Turkey
Information: dumitru@cankaya.edu.tr;
http://www.cankaya.edu.tr/fda08/
17–21: Geometry and topology in low dimension, CIRM
Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
24–28: Approximation, geometric modelling and applications,
CIRM Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
22–26: 10th International workshop in set theory, CIRM
Luminy, Marseille, France
Information: colloque@cirm.univ-mrs.fr;
http://www.cirm.univ-mrs.fr
EMS Newsletter December 2007 51

New books published by the
20% discount for individual members
of the European, American, Australian and
Canadian Mathematical Societies!
Guus Balkema (University of Amsterdam, The Netherlands), Paul Embrechts (ETH Zurich, Switzerland)
High Risk Scenarios and Extremes – A Geometric Approach (Zurich Lectures in Advanced Mathematics)
ISBN 978-3-03719-035-7. 2007. 388 pages. Softcover. 17.0 cm x 24.0 cm. 48.00 Euro
Quantitative Risk Management (QRM) has become a fi eld of research of considerable importance to numerous areas of application, including insurance,
banking, energy, medicine, reliability. Mainly motivated by examples from insurance and fi nance, the authors develop a theory for handling
multivariate extremes. The approach borrows ideas from portfolio theory and aims at an intuitive approach in the spirit of the Peaks over Thresholds
method. The book is based on a graduate course on point processes and extremes and primarily aimed at students in statistics and fi nance as well
as professionals involved in risk analysis.
Dorothee D. Haroske (University of Jena, Germany), Hans Triebel (University of Jena, Germany)
Distributions, Sobolev spaces, Elliptic equations (EMS Textbooks in Mathematics)
ISBN 978-3-03719-042-5. 2007. 303 pages. Hardcover.16.5 cm x 23.5 cm. 48.00 Euro
It is the main aim of this book to develop at an accessible, moderate level an L 2
theory for elliptic differential operators of second order on bounded
smooth domains in Euclidean n-space, including a priori estimates for boundary-value problems in terms of (fractional) Sobolev spaces on domains
and on their boundaries, together with a related spectral theory. The presentation is preceded by an introduction to the classical theory for the
Laplace–Poisson equation, and some chapters providing required ingredients such as the theory of distributions, Sobolev spaces and the spectral
theory in Hilbert spaces.
Gohar Harutyunyan (University of Oldenburg, Germany), B.-Wolfgang Schulze (University of Potsdam, Germany)
Elliptic Mixed, Transmission and Singular Crack Problems (EMS Tracts in Mathematics Vol. 4)
ISBN 978-3-03719-040-1. 2007. 777 pages. Hardcover. 17.0 cm x 24.0 cm. 112.00 Euro
Mixed, transmission, or crack problems belong to the analysis of boundary value problems on manifolds with singularities. The Zaremba problem
with a jump between Dirichlet and Neumann conditions along an interface on the boundary is a classical example. The central theme of this book is
to study mixed problems in standard Sobolev spaces as well as in weighted edge spaces where the interfaces are interpreted as edges. Parametrices
and regularity of solutions are obtained within a systematic calculus of boundary value problems on manifolds with conical or edge singularities.
Ralf Meyer (University of Göttingen, Germany)
Local and Analytic Cyclic Homology (EMS Tracts in Mathematics Vol. 3)
ISBN 978-3-03719-039-5. 2007. 368 pages. Hardcover. 17.0 cm x 24.0 cm. 58.00 Euro
Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham
cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as
most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras.
In this book the author develops and compares these theories, emphasising their homological properties.
Sergei Buyalo (Steklov Institute of Mathematics, St. Petersburg, Russia), Viktor Schroeder (University of Zurich, Switzerland)
Elements of Asymptotic Geometry (EMS Monographs in Mathematics)
ISBN 978-3-03719-036-4. 2007. 212 pages. Hardcover. 16.5 cm x 23.5 cm. 58.00 Euro
Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important
class of model spaces are the hyperbolic spaces (in the sense of Gromov), for which the asymptotic geometry is nicely encoded in the boundary at
infi nity. In the fi rst part of this book, in analogy with the concepts of classical hyperbolic geometry, the authors provide a systematic account of the
basic theory of Gromov hyperbolic spaces. In the second part of the book, various aspects of the asymptotic geometry of arbitrary metric spaces
are considered.
Handbook of Teichmüller Theory, Volume I (IRMA Lectures in Mathematics and Theoretical Physics)
Athanase Papadopoulos (IRMA, Strasbourg, France), Editor
ISBN 978-3-03719-029-6. 2007. 802 pages. Hardcover. 17.0 cm x 24.0 cm. 98.00 Euro
The Teichmüller space of a surface was introduced by O. Teichmüller in the 1930s. It is a basic tool in the study of Riemann‘s moduli space and of
the mapping class group. These objects are fundamental in several fi elds of mathematics including algebraic geometry, number theory, topology,
geometry, and dynamics. The purpose of this handbook is to give a panorama of some of the most important aspects of Teichmüller theory. The
handbook will be useful to specialists in the fi eld, to graduate students, and more generally to mathematicians who want to learn about the subject.
All the chapters are self-contained and have a pedagogical character. They are written by leading experts in the subject.
Andrzej Schinzel, Selecta (Heritage of European Mathematics)
Volume I: Diophantine Problems and Polynomials, Volume II: Elementary, Analytic and Geometric Number Theory
Henryk Iwaniec (Rutgers University, USA). Władysław Narkiewicz (University of Wrocław, Poland), Jerzy Urbanowicz (IM PAN, Warsaw, Poland), Editors
ISBN 978-3-03719-038-8. 1417 pages. Hardcover. 17.0 cm x 24.0 cm. 168.00 Euro
These Selecta contain Andrzej Schinzel‘s most important articles published between 1955 and 2006. The arrangement is by topic, with each major
category introduced by an expert‘s comment.
European Mathematical Society Publishing House
Seminar for Applied Mathematics, ETH-Zentrum FLI C4
Fliederstrasse 23
CH-8092 Zürich, Switzerland
orders@ems-ph.org
www.ems-ph.org

Recent Books
edited by Ivan Netuka and Vladimír Souček (Prague)
Books submitted for review should be sent to: Ivan Netuka,
MÚUK, Sokolovská, 83, 186 75 Praha 8, Czech Republic.
S. Alinhac, P. Gérard: Opérateurs pseudo-différentiels et
théorème de Nash-Moser, Mathématiques, EDP Sciences, Les
Ulis, 2000, 188 pp., EUR 35, ISBN 2-7296-0364-6, ISBN 2-222-
04535-7
This book is a course on partial differential equations and pseudodifferential
operators. It consists of three parts. The choice of
material together with its arrangement shows some non-obvious
links between seemingly distant topics of interest. The first
part develops the theory of pseudodifferential operators. This
generalization of partial differential operators is based on the
fact that differentiation acts as polynomial multiplication in
the Fourier regime. If we use a (reasonable) non-polynomial
function as the multiplier, we obtain a pseudodifferential operator.
The multiplier may depend on the initial space variable.
Following this the case of variable coefficients is covered. The
objective here is to establish the most important formulas of
the ensuing calculus.
The second part presents the Littlewood-Paley theory and
microlocal analysis (in particular a concept of the wave front
set of a distribution in connection with pseudodifferential operators,
energy estimates and propagation of singularities). In
comparison with the first chapter, the exposition moves towards
more specific problems motivated by partial differential
equations but still in the pseudodifferential setting.
The third part studies perturbations of problems in partial
differential equations. Starting with applications of the implicit
function theorem and going through the situation treated by
fixed point theorems, the main goal of the chapter is the Nash-
Moser theorem. The results describe existence problems and estimates
for solutions of perturbed problems. The main difficulty
is to handle the loss of derivatives appearing when solving the
linearized problem. The Nash theorem on existence of an isometric
embedding of a Riemannian manifold is included as a
special case. The book is intended as a course for advanced students
but it will also be very useful for researchers. The material
contained here is deep and very important for the understanding
of some issues of the theory of partial differential equations and
the more general context of pseudodifferential operators. The
presentation is quite compact and the student should be well
prepared (in particular, a good knowledge of Fourier calculus
is needed). When the authors say “elementary” it should sometimes
be read as “short”. The exposition is highly self-contained
and the text is complemented by numerous exercises. (jama)
G. Allaire: Numerical Analysis and Optimization. An introduction
to mathematical modelling and numerical simulation,
Numerical Mathematics and Scientific Computation,
Oxford University Press, Oxford, 2007, 455 pp., GBP 30, ISBN
978-0-19-920522-6
This book represents a modern introduction to the numerical
analysis of partial differential equations and to optimization
Recent books
techniques. The goal is to introduce the reader to the world
of mathematical modelling and numerical simulation. It contains
finite difference as well as finite element methods for
numerical solution of stationary and non-stationary problems.
Moreover, the book also treats optimization and operational
research techniques. The first chapter introduces the reader to
the area of mathematical modelling and numerical simulation.
It contains examples of problems leading to partial differential
equations of various types and explains some basic questions
and obstacles met in their numerical solution. Chapter
2 is concerned with the finite difference method. In chapter 3
the variational formulation of stationary boundary value problems
is introduced. Chapter 4 is devoted to the explanation of
the concept and basic properties of Sobolev spaces. Chapter 5
presents the qualitative theory of elliptic problems. Chapters
3 to 5 form the basis for the treatment of the finite element
method in chapter 6.
Chapter 7 is concerned with eigenvalue problems. In chapter
8, basic qualitative properties and the numerical solution of
evolution problems are treated. Chapters 9 to 11 are devoted
to optimization techniques. In these chapters, the motivation
and examples are given and optimality conditions and basic optimization
algorithms are discussed. Finally, methods of operational
research are presented. The book contains two appendices:
‘Review of Hilbert spaces’ and ‘Matrix numerical analysis’.
The book represents an interesting modern treatment of classical
numerical techniques. It contains a number of examples
accompanied by results of computations and figures showing
solutions of various problems. A positive feature of the book
is the fact that it needs only facts from the first few years of
university study. Therefore, it can be recommended to students
of mathematics and engineering sciences, applied and numerical
mathematicians, engineers, physicists and specialists dealing
with modelling and numerical simulation of various structures
and processes. (mf)
A.C. Atkinson, A.N. Donev, R.D. Tobias: Optimum Experimental
Designs, with SAS, Oxford Statistical Science Series 34,
Oxford University Press, Oxford, 2007, 511 pp., GBP 35, ISBN
978-0-19-929660-6
The purpose of this book is to describe a sufficient amount of
the theory to make apparent the overall pattern of optimum
designs and to provide a thorough background for the use of
packages of SAS software in the design of optimum experiments.
The book is divided into two parts. The idea of the statistical
design of experiments is introduced in the first shorter
part. The theory of an optimum experimental design is developed
in the second part together with many applications, procedures
and examples. The first part (Background) consists of
eight chapters. The authors discuss advantages of a statistical
approach for the design of experiments and introduce many
models and examples, which are applied mainly in the second
part of the book. The first part describes stages in experimental
research, the choice of model, the least squares method, a design
matrix, standard design and analysis of experiments. The
packages for data processing and statistical analysis (SAS software)
are mentioned here and these packages are used in many
examples in the second part (Theory and Applications).
The second part starts with a formulation of the general
equivalence theorem. Then follows the criteria of A-, D- and E-
EMS Newsletter December 2007 53

Recent books
optimality, algorithms for the construction of exact D-optimum
design and optimum experimental design with SAS. Chapter
14 is concerned with the design of experiments, where the response
depends on both qualitative and quantitative factors.
Mixture experiments are described in chapter 15. The comprehensive
chapter 17 describes an extension of previous techniques
to nonlinear regression models, including those defined
by systems of differential equations. The following chapters
contain a discussion of Bayesian optimum designs, augmented
design, model checking and design of discriminating between
models, compound design criteria, generalized linear models,
response transformation and structured variances, time–dependent
models with correlated observations, design of clinical
trials and exercises. The whole book is oriented to the practical
mind with many examples and procedures. The examples are
mainly drawn from scientific, engineering, pharmaceutical and
agriculture practice. The book contains a large list of references.
It can be recommended mainly to scientists and to students
preparing a Masters or a doctoral thesis. (jzit)
D. Busneag: Categories of Algebraic Logic, Editura Academiei
Romane, Bucurest, 2006, 270 pp., ISBN 978-973-27-1381-5
This monograph summarizes the author’s contributions to the
study of algebras having their origin in logic. The book is divided
into five chapters. The first four chapters (sets and functions,
ordered sets, topics in universal algebra and topics in theory
of categories) contain basic material, which is needed for the
main part (algebras of logic). The corresponding chapter deals
with several categories of algebras coming from intuitionistic
logic, logic without contraction and fuzzy logic, in particular
with Heyting algebras, Hilbert algebras, Hertz algebras, residuated
lattices and Wajsberg algebras. The author concentrates on
the algebraic and categorical properties of these structures contained
in his numerous research articles. Historical background
and motivation for the results described are not included; the
reader is referred to the literature. (lbar)
I. Chiswell, W. Hodges: Mathematical Logic, Oxford Texts in
Logic 3, Oxford University Press, Oxford, 2007, 250 pp., GBP
29.50, ISBN 978-0-19-921562-1
This book gives a standard first course in propositional and
predicate logic. No prior knowledge of the subject is needed
to read the book. The basic material is concentrated in chapter
3 (propositional logic), chapter 5 (quantifier-free logic)
and chapter 7 (first-order logic, including completeness and
compactness). Chapter 8 contains deeper results, in particular
Church undecidability, Gödel undecidability and the incompleteness
theorems. The proofs are based on Matiyasevich’s
theorem, formulated in 5.8.6 using the notions of computable
and computably enumerable sets stated in 5.8.4 on an intuitive
(but acceptable) level. Chapters 1, 2, 4 and 6 contain the
motivation and a discussion of formal and psychological aspects
of logic considerations in applications. Many examples
and exercises are included. Moreover, appendix B provides
some information on denotational semantics. The book is written
on the foundation of the extensive teaching experience of
the authors. They have succeeded in giving a presentation of
the subject in a formally correct and intuitively acceptable way.
They also include some applications in computer science and
linguistics. (jmlc)
D. Christodoulou: The Formation of Shocks in 3-Dimensional
Fluids, EMS Monographs in Mathematics, European Mathematical
Society, Zürich, 2007, 992 pp., EUR 148, ISBN 978-3-
03719-031-9
Dealing with the relativistic Euler equations for an ideal fluid
with an arbitrary equation of state, this book provides an original,
mathematically sound and complete theory of the formation
of shocks in a three-dimensional spatial setting. More precisely,
starting with initial data that coincide outside of a sphere
with the data corresponding to a constant state, and assuming a
suitable restriction on the size of an initial perturbation of the
constant state, theorems describing maximal classical development
are established. It is shown, in particular, that the boundary
of the maximal classical development includes a singular part,
where the inverse density of the wave fronts vanishes, which indicates
a formation of shocks. The central concept used in the
book is an acoustic spacetime manifold. Methods from differential
geometry play an important role in the book. (jmal)
S. Crovisier, J. Franks, J.-M. Gambaudo, P. Le Calvez: Dynamiques
des difféomorphismes conservatifs des surfaces:
un point de vue topologique, Panoramas et Synthèses, no. 21,
Société Mathématique de France, Paris, 2006, 143 pp., EUR 36,
ISBN 978-2-85629-220-4
This book contains a written version of lectures presented at
the meeting of the Société Mathématique de France held in
the summer of 2004 at Dijon. It consists of four contributions.
The first one (by S. Crovisier) is devoted to a description of the
space of orbits (and in particular the space of periodic orbits) of
a chosen volume preserving diffeomorphism f of a smooth compact
connected surface with a given smooth volume. To avoid
pathological cases, the author considers C1-generic surface diffeomorphisms.
The next contribution (by J. Franks) studies distortion
elements in groups of surface diffeomorphisms (and, for
comparison, also for circle diffeomorphisms). The results are
applied to group actions on surfaces preserving a Borel measure.
Three different topics and their relations are treated in the
third contribution (by J.-M. Gambaudo): topological invariants
associated with flows on three-dimensional manifolds, the
space of configurations of an incompressible fluid on an oriented
manifold, and a structure of the group of diffeomorphisms
(preserving a given area form and isotopic to the identity) of
a compact oriented surface. Finally, P. Le Calvez studies in the
last contribution various versions of the Brouwer plane translation
theorem and its relation to the study of homeomorphisms
of surfaces (including a proof of the Conley conjecture in the
case of a compact surface of genus greater than zero). (vs)
W. Ebeling: Functions of Several Complex Variables and their
Singularities, Graduate Studies in Mathematics, vol. 83, American
Mathematical Society, Providence, 2007, 312 pp., USD 59,
ISBN 978-0-8218-3319-3
This book belongs to the AMS series ‘Graduate Studies in
Mathematics’. Its main topic is a study of isolated singularities
of hypersurfaces in several complex variables. The first chapter
describes the classical theory of Riemann surfaces (coverings,
fundamental groups, branched meromorphic continuation, the
Riemann surfaces of an algebraic function and the Puiseux expansion).
Basic facts of the theory of functions of several complex
variables are explained in chapter 2 (the implicit function
54 EMS Newsletter December 2007

Recent books
theorem, the Weierstrass preparation theorem and its generalizations,
germs of analytic sets and their dimensions and special
cases of the Remmert mapping theorem). Chapter 3 includes
basic facts from differential geometry (smooth manifolds and
their tangent bundles, transversality, homogeneous spaces and
complex manifolds) together with a study of isolated critical
points of holomorphic functions (the complex Morse lemma,
universal unfolding, morsification and the classification of simple
singularities).
Chapter 4 contains a description of basic facts from differential
topology (the Ehresmann fibration theorem, a holonomy
group of a fiber bundle, singular homology groups, the Euler
characteristic, intersection and linking numbers, the braid
group and the homotopy sequence of a fiber bundle). The last
chapter is devoted to a description (partly without proofs) of
topological properties of isolated critical points of holomorphic
functions (including a discussion of monodromy groups
and vanishing cycles, the Picard-Lefschetz theorem, the Milnor
fibration, the intersection matrix and the Coxeter-Dynkin diagrams,
variation of singularities, action of the braid group and
the Arnold classification of unimodal singularities). The book
contains a lot of illustrative pictures and diagrams substantially
helping the geometric intuition of the reader. (vs)
E.A. Fellmann: Leonard Euler, Birkhäuser, Basel, 2007, 179 pp.,
EUR 31,99, ISBN 978-3-7643-7538-6
In spite of the existence of many texts devoted to the description
of Euler’s life and in spite of the fact that this is “a book
about a great mathematician, which is entirely formulae-free”,
the book brings many facts of interest even for a professional
mathematician (as was the intention of the original publisher).
The author follows Euler’s career in detail. The first chapter contains
a brief copy of Euler’s curriculum vitae (in German, with
English translation, dictated by Euler to his son in 1767) written
about 17 years before his death. It also contains a description of
Euler’s Basel period. Other chapters contain an account of the
first Petersburg stay, the Berlin period and the second stay in
Petersburg. A reader will learn facts about Euler’s parents, his
close friends, his children and both his marriages as well as many
fields influenced by Euler’s mathematical work. The book contains
nice pictures, ten pages of bibliographical sources and a lot
of notes. The book will be of interest to everybody who wants to
know more about the unique phenomenon of ”Euler”, the most
productive mathematician in the history of mathematics. (jive)
U. Grenander, M. Miller: Pattern Theory. From Representation
to Inference, Oxford University Press, Oxford, 2006, 596
pp., GBP 100, ISBN 0-19-850570-1
This book summarizes Grenander’s lifelong efforts to represent
empirical knowledge of complex real world processes in
a mathematical form. The authors characterize the book’s aim
as “the formalization of a small set of ideas for constructing the
representation of the pattern themselves, which accommodate
variability and structure simultaneously”. After a short introduction,
the basic paradigm of Bayesian setup of the estimation
on the posterior distribution is explained in chapter 2. The
next four chapters analyze the role of pattern representations
in conditioning structure. Chapters 7 and 8 examine groups of
geometric transformations suitable for the representation of
geometric objects. Chapter 9 is devoted to random processes
and fields on the background spaces – the continuum limits of
the finite graphs (e.g. Gaussian processes representing physical
processes in the world). Chapters 10 and 11 develop metric
space structure of shapes and further extensions of this approach
are treated in the next chapter.
Chapters 13 to 15 pass from pure representations of shape
to their Bayes estimations and parametric representations. The
next two chapters are devoted to infinite dimensional shape covered
by a new field of computational anatomy, an emerging discipline
introduced by Miller and Grenander at the end of the last
century with applications in the biological variability of human
anatomy. The last two chapters conclude the inference under the
assumption that the posterior distributions describing patterns
contain all the information about the underlying regular structure.
Markov processes are introduced as models for changes
of this structure and random samples are generated via their
simulation. The book is designed for a broad public; it contains
numerous exercises, examples of applications and an extended
bibliography. Appendices outlining some theorems, their proofs
and solutions of selected problems are on the website: www.oup.
com/uk/booksites/content/0198505701/appendices. (is)
A. Ya. Helemskii: Lectures and Exercises on Functional Analysis,
Translations of Mathematical Monographs, vol. 233, American
Mathematical Society, Providence, 2006, 468 pp., USD 129,
ISBN 978-0-8218-4098-6
This book on functional analysis is written with a considerable
use of the language of category theory. The introductory chapter
is devoted to foundations (metric and topological spaces,
categories, morphisms and functors). The next chapters cover
basic elements of functional analysis: normed spaces and operators
(and the Hahn-Banach theorem and an introduction
to quantum functional analysis) and Banach spaces (including
tensor products). A special chapter presents the theory of polynormed
spaces (basically locally convex spaces), weak topologies
and the theory of distributions. Further chapters are devoted
to compact and Fredholm operators and spectral theory
(among others C*-algebras and Borel functional calculus). The
final chapters deal with Fourier transforms and fundamental
harmonic analysis on groups. The book contains many examples
and exercises of different levels of difficulty. It requires only a
basic knowledge of linear algebra, elements of real analysis, and
metric spaces. It can be recommended to a broad spectrum of
readers, to graduate students as well as young researchers. (jl)
B. Helffer, F. Nier: Quantitative Analysis of Metastability in
Reversible Diffusion Processes Via a Witten Complex Approach
– The Case with Boundary, Mémoires de la Société
Mathématique de France, no. 105, Société Mathématique de
France, Paris, 2006, 89 pp., EUR 26, ISBN 978-2-85629-218-1
In the 80s, E. Witten introduced a version of the Laplacian on
p-forms, distorted by a function f. If a Morse function f is given
on M = R n (or on a compact Riemannian manifold M without
a boundary) and if h is a positive constant, an analysis of the
properties of small eigenvalues of the semiclassical Witten Laplacian
∆(f,h) was given by A. Bovier, M. Eckhoff,V. Gayrard,
B. Helffer, M. Klein and F. Nier. The main theme of the book
presented here is to extend the mentioned results to the case
with boundary (a domain in R n and its smooth boundary, or
a compact connected oriented Riemannian manifold with a
EMS Newsletter December 2007 55

Recent books
boundary) for 0-forms. An important part of the work consists
of a construction of a Witten cohomology complex adapted to
the case with boundary. (vs)
G. Helmberg: Getting acquainted with fractals, Walter de Gruyter,
Berlin, 2007, 177 pp, EUR 78, ISBN 978-3-11-019092-2
This book provides an answer to the question of what type of
mathematics is behind the famous fractal pictures. One of the
main features of fractal objects is their (often non-integral)
dimension. The first chapter presents the foundations of the
theory of Hausdorff measure and dimension; it shows ways to
produce fractal sets by means of “deleting and replacing” and
it gives ideas of how to compute the dimension of such objects.
The second chapter is related to the second main feature of
fractals, namely their self-similarity. Iterative function systems
are studied here. Starting with a finite number of similarities, by
iteration we obtain a fractal set as the limit image. A rigorous
theoretical treatment is complemented by many representative
examples showing how a proper choice of iterated mappings
with carefully chosen parameters may lead to nice pictures (including
island archipelago, tree, Barnsley fern and grass).
In the third chapter, the theory of iteration of complex polynomials,
resulting in Julia sets and the Mandelbrot set, is developed.
Again, examples of impressive pictures based on precise
formulas illustrate the theory. The exposition is written as a
rigorous mathematical text with precise definitions, theorems
and some proofs. It is readable with skills at the level of an undergraduate
student. However, it is not intended as a textbook.
In particular, the presentation is not self-contained and difficult
proofs are referred to. For readers interested in the full depth of
the theory it is recommended to supplement their studies with
other sources. The purpose of this book is just to yield a bridge
between the fractal pictures and serious mathematics and this
is successfully achieved. (jama)
D.D. Joyce: Riemannian Holonomy Groups and Calibrated
Geometry, Oxford Graduate Texts in Mathematics 12, Oxford
University Press, Oxford, 2007, 303 pp., GBP 34.50, ISBN 978-
0-19-921559-1
The main topic treated in this book is the classical subject of
Riemannian holonomy groups, together with a new theme of
so-called calibrated submanifolds. The book is designed for
graduate students of mathematics (and theoretical physics) and
it introduces the reader to modern topics important for understanding
new mathematics emerging from string theory.
The first part of the book is devoted to an explanation of
holonomy groups in a Riemannian setting (including a short review
of background material, connections, curvature, holonomy
groups, G-structures, Riemannian symmetric spaces, the classification
of Riemannian holonomy groups and Kähler manifolds).
Special chapters are devoted to the Calabi conjecture
and its proof, Calabi-Yau manifolds and their deformations,
hyperkähler and quaternionic Kähler manifolds and the exceptional
holonomy groups. Most of this material is related to a
previous important monograph of the author, ‘Compact manifolds
with special holonomy, Oxford University Press, 2000’.
Completely new material is contained in chapters on calibrated
submanifolds, special Lagrangian geometry, mirror symmetry
and the Strominger-Yau-Zaslow conjecture. The book ends
with a chapter on special types of calibrated submanifolds of
manifolds with exceptional holonomy. The book is very wellwritten,
it contains a wealth of important material and it should
be on the shelf of everybody interested in the topic. (vs)
M. Junge, Ch. Le Merdy, Q. Xu: H ∞ Functional Calculus and
Square Functions on Noncommutative L p -Spaces, Astérisque,
no. 305, Société Mathématique de France, Paris, 2006, 138 pp.,
EUR 26, ISBN 978-2-85629-189-4
H ∞ functional calculus, introduced by Alan McIntosh in the 80s
and developed in collaboration with M. Cowling, I. Doust and
A. Yagi, plays an important role in several branches of operator
theory. The main topic treated in the book is natural square functions
associated with sectorial operators, or with semigroups,
acting on suitable non-commutative L p -spaces and their relations
with H ∞ functional calculus. After providing a background
on non-commutative L p -spaces, H ∞ functional calculus, sectorial
operators and semigroups, the authors introduce square
functions and describe their relations to H ∞ functional calculus.
There is a chapter devoted to a noncommutative generalization
of the Stein diffusion semigroup. Further topics include multiplication
operators, Hamiltonians and the Schur multipliers
on Schatten space, semigroups on q-deformed von Neumann
algebras, and the noncommutative Poisson semigroup of a free
group. The last chapter briefly treats the non tracial case. (vs)
P. Kaye, R. Laflamme, M. Mosca: An Introduction to Quantum
Computing, Oxford University Press, Oxford, 2006, 274 pp.,
GBP 75, ISBN 0-19-857000-7
The area of quantum computation and quantum algorithms is
developing very quickly. This book describes very carefully and
understandably its foundations. The book is intended for a broad
spectrum of possible readers from various fields (including physicists
and engineers) and it requires only a modest knowledge
(basic facts of linear algebra). More advanced topics (e.g. tensor
products and spectral properties) are carefully reviewed. The
authors treat in a systematic way the main topics in the field.
This starts with a review of basic models of computation (going
from classical to quantum). After a careful review of notions of
linear algebra (and Dirac notation), they explain basic axioms
of quantum mechanics and basic models for quantum computations.
After a description of basic protocols for quantum information
(superdense coding and quantum teleportation), they
concentrate on quantum algorithms. They discuss many of these
(including the Shor algorithm and the Gover quantum search algorithm).
The last two chapters are devoted to quantum computation
complexity theory and quantum error corrections. There
are a lot of examples throughout the text and the treatment of
all topics included in the book is very understandable, clear and
systematic. There are no doubts that the book will be very useful
for students from various branches of science. (vs)
D. Khoshnevisan: Probability, Graduate Studies in Mathematics,
vol. 80, American Mathematical Society, Providence, 2007,
224 pp., USD 45, ISBN 978-0-8218-4215-7
This book presents a graduate course in measure-theoretic
probability designed to cover a one year course at
a comfortable pace. The author starts with classical probability
(including a neat proof of the de Moivre-Laplace
theorem) before moving on to elements of measure theory
(measure spaces, integration, modes of convergence
56 EMS Newsletter December 2007

Recent books
and product spaces). The chapter on independence covers
the strong law of large numbers and the Glivenko-
Cantalli theorem. The central limit theorem in an independent
and identically distributed setting is proved
both by the classical harmonic-analytic approach and by
the Liapunov-Lindeberg replacement method. The material
also covers elements of weak convergence and the
Cramer characterization of normal distributions.
Some rich material on conditional expectations and discrete
time martingales is offered, the optional stopping theorem and
the martingale convergence theorem being the principal results.
The applications range from the Lebesgue differentiation theorem
to option-pricing procedures. Brownian motion is constructed,
its nowhere differentiability and the strong Markov property
being the principal achievements. The text terminates with a short
introduction to stochastic calculus (the Itô integral and formula,
optional stopping, L2-martingales and the exit distribution of
Brownian motion). The reader might miss a treatment of Markov
chains but overall the text is neatly written and presents not only
good mathematics but also a heuristic view of modern probability,
including numerous exercises and historical notes. (jstep)
J. M. Lee: Fredholm operators and Einstein metrics on conformally
compact manifolds, Memoirs of AMS, no. 864, American
Mathematical Society, Providence, 2006, 83 pp., USD 55, ISBN
978-0-8218-3915-7
Conformal geometry (in particular in dimension 4) and corresponding
geometric partial differential equations have been
studied carefully in the last few decades. New tools developed
recently by Ch. Fefferman, R. Graham, M. Eastwood, R. Gover
and T. Bailey have allowed a much better understanding of
conformal invariants (including a complete classification in
particular cases). A principal tool here is the ambient metric
construction of Fefferman and Graham. The so-called AdS-
CFT correspondence recently emerging from string theory is
based on the notion of conformal infinity for a (pseudo-)Riemannian
metric developed originally by R. Penrose. It caused a
dramatic rise of interest in these problems both in mathematics
and mathematical physics. The problem treated in the book is
to find, for a compact manifold M with a given conformal structure
c on its boundary ∂M, a complete asymptotically hyperbolic
Einstein metric g on the interior of M such that c coincides
with the conformal class of g on ∂M.
The book contains a systematic and self-contained description
of the perturbation version of the problem (where we are
looking for solutions of the problem for c near to a given c 0
, for
which the solution exists). An important fact is that the solution
has an optimal Hölder regularity up to the boundary (for
n even). Related results (with less boundary regularity but also
for the quaternionic and octonionic cases) were described recently
in the book by O. Biquard (see EMS Newsletter 65, page
55). Methods used in the book are based on general sharp Fredholm
and isomorphism theorems for geometric (degenerate)
linear elliptic operators. (vs)
Q. Liu: Algebraic Geometry and Arithmetic Curves, Oxford
Graduate Texts in Mathematics 6, Oxford University Press, Oxford,
2006, 577 pp., GBP 34.50, ISBN 0-19-920249-4
The main topics of this book are arithmetic surfaces and reductions
of algebraic curves. But these topics are systematically
treated only in chapters 8, 9, and 10. The aim of the first seven
chapters is to educate the reader in the theory of schemes. The
author devotes the whole first chapter to commutative algebra.
Nevertheless, one chapter on algebra is not sufficient for the
purposes of algebraic geometry and the author has to make digressions
and introduce further algebraic notions and results
several times. (It is interesting to mention that the author had
the idea to present algebraic aspects in a simpler form; he succeeded
in avoiding homological algebra.) Thanks to these algebraic
parts, the book is rather self-contained.
Next the theory of schemes is studied. This presentation is
excellent indeed. It is not a survey but a systematic and thorough
theory of schemes covering many aspects of them. In spite
of many necessary details, the text reads very well and moreover,
we are not lost in this rich theory and it is possible to keep
a certain global overview. The author includes a substantial
number of examples, which is very helpful for the reader. Every
section is followed by a long series of exercises. They are of two
kinds. Some of them are designed to increase understanding of
the text, the others extend and complete the theory. These seven
chapters can be strongly recommended as a basis for a course
on the theory of schemes. I agree with the author that even a
good undergraduate student can learn a lot from this book.
The second part uses techniques developed in the first seven
chapters. It begins with a description of blowing-ups. Then fibered
surfaces over a Dedekind ring and desingularizations of surfaces
are studied. The next topic is intersection theory on an arithmetic
surface. The last chapter is devoted to the reduction theory
of algebraic curves. Special attention is paid to elliptic curves. At
the end, stable curves and stable reductions are treated; in particular,
the Artin-Winter proof of the stable reduction theorem
of Deligne-Mumford. Some concrete computations are also presented.
These last chapters introduce the reader to contemporary
research. He or she will also find hints for further reading. (jiva)
G. Maltsiniotis: La théorie de l’homotopie de Grothendieck,
Astérisque 301, Société Mathématique de France, Paris, 2005, 140
pp., EUR 26, ISBN 2-85629-181-3
D.-C. Cisinski: Les préfaisceaux comme modeles des types
d’homotopie, Astérisque 308, Société Mathématique de France,
Paris, 2006, 390 pp., EUR 78, ISBN 978-2-85629-225-9
The aim of the book by G. Maltsiniotis is to give an introduction
to some of the key ideas of A. Grothendieck’s unpublished treatise
on abstract homotopy theory via category theory (“Pursuing
Stacks”). The fundamental concept is that of a “test category”
A, whose main property is that the localization of the presheaf
category à with respect to weak equivalences is naturally equivalent
to the usual homotopy category (a typical example is the
category of simplices/cubes, for which à is the category of simplicial/cubical
sets). Furthermore, one can consider more general
homotopy categories by replacing the class of weak equivalences
by an arbitrary “fundamental localize” (a class of functors that
satisfies, among other things, Quillen’s Theorem A). The exposition
of these concepts and of their basic properties is very clear.
The book by Cisinski develops Grothendieck’s theory further.
One of its main themes is the question of existence of a well-behaved
closed model structure on à as a characterization of (accessible)
test categories A, as conjectured by Grothendieck. Other
topics include a discussion of equivariant homotopy and examples
of interesting test categories and homotopy categories. (jnek)
EMS Newsletter December 2007 57

Recent books
C. Mazza, V. Voevodsky, C. Weibel: Lecture Notes on Motivic
Cohomology, Clay Mathematics Monographs, vol. 2, American
Mathematical Society, Providence, 2006, 216 pp., USD 45, ISBN
978-0-8218-3847-1
This book by C. Mazza and Ch. Weibel is based on a series of
lectures given by V. Voevodsky in Princeton in the academic
year 1999/2000. The idea of these lectures is to relate motivic
cohomology to other known invariants of algebraic varieties
and rings. The power of motivic cohomology as a tool for
proving results in algebra and algebraic geometry lies in the
interaction with properties of motivic cohomology itself – its
homotopy invariance, Mayer-Vietoris and Gysin long exact sequences,
projective bundles, etc. The contents of the book may
be divided into two parts, corresponding to the fall and spring
terms. The fall term lectures contain the definition of motivic
cohomology and proofs for various comparison results (e.g.
with the Milnor K-group and the Picard group). The spring
term lectures include more advanced results in the theory of
sheaves with transfers and a proof of the general comparison
result with higher Chow groups of algebraic varieties. (pso)
C. Meyer: Modular Calabi-Yau Threefolds, Fields Institute
Monographs, American Mathematical Society, Providence, 2005,
194 pp., USD 59, ISBN 0-8218-3908-X
The Taniyama-Shimura conjecture (used in the proof of Fermat’s
last theorem) relates the number of points on an elliptic
curve over a finite field to Fourier coefficients of a modular
form of weight two. Calabi-Yau manifolds are generalizations
of elliptic curves to higher dimensions and they play a prominent
role in the contemporary development of string theory.
Their arithmetic properties in dimensions two (i.e. properties of
K3 surfaces) have been studied recently. This book is devoted
to the case of Calabi-Yau manifolds in dimension three. The
dimension of the middle étale cohomology of a Calabi-Yau
three-fold M gives key information on its arithmetic properties.
If it equals two, the three-fold is called rigid. There is a precise
conjecture on a connection of a rigid three-fold with modular
forms, whereas the situation is much more complicated for nonrigid
ones. The book contains hundreds of examples of rigid
and non-rigid cases. The first chapter reviews basic facts on
Calabi-Yau manifolds and their arithmetic. The next four chapters
contain a discussion of a considerable number of explicit
examples (including many different quintics, double octics and
various complete intersections). In the last chapter, the author
describes the correspondences between Calabi-Yau three-folds
having the same modular form in their L-series. The book is
based on the author’s PhD thesis. (vs)
A. Moroianu: Lectures on Kähler Geometry, London Mathematical
Society Student Texts 69, Cambridge University Press,
Cambridge, 2007, 171 pp., GBP 19.99, ISBN 978-0-521-68897-0
This book contains a written version of lectures on Kähler
geometry prepared for graduate students of mathematics
and theoretical physics from a perspective of Riemannian geometry.
The first part explains basic notions from differential
geometry (manifolds, tensor fields, integration of differential
forms, principal and vector bundles, connections, Riemannian
manifolds, parallel transport and a concept of holonomy). The
second part is devoted to complex and Hermitean geometry
(complex manifolds, complex and holomorphic vector bundles,
Hermitean vector bundles, the Chern connection, Kähler metrics,
the Kählerian curvature tensor, the Laplace operator on a
Kähler manifold, Hodge theory and Dolbeault theory).
Special and more advanced topics are discussed in the
last part of the book, including a short description of the first
Chern classes of vector bundles, Ricci-flat Kähler manifolds,
the Calabi-Yau theorem, the Aubin-Yau theorem on Kähler-
Einstein metrics, vanishing theorems on Kähler manifolds, the
Hirzebruch-Riemann-Roch formula, hyperkähler manifolds
and Calabi-Yau manifolds). Recent intensive collaboration between
mathematics and theoretical physics in areas connected
with string theory has, as a consequence, created substantially
stronger requirements on the mathematical background of
young people aiming to work in this broad field. The book covers
in a nice, systematic and understandable way basic tools of
differential geometry. It can be useful both for student readers
and for teachers preparing lectures on the subject. (vs)
J. Nagel, C. Peters: Algebraic Cycles and Motives, vol. 2, London
Mathematical Society Lecture Notes Series 344, Cambridge University
Press, Cambridge, 2007, 359 pp., ISBN 978-0-521-70175-4
The second volume of the proceedings of J. Murre’s 75th birthday
conference (held in Leiden in 2004) contains fifteen research
articles on various aspects of motives and algebraic cycles.
The two main themes represented here are the conjectures
of Bloch and Beilinson (refined by Murre) on the existence of
a canonical filtration on Chow groups (articles by Beauville;
Bloch and Esnault; Jannsen; Kahn, Murre and Pedrini; and
Miller, Müller-Stach, Wortmann, Yang and Zuo) and Hodgetheoretical
methods in the study of algebraic cycles (articles by
Asakura and S. Saito; Lewis; Nagel; Peters and Steenbrink; and
M. Saito). Other topics include: finite-dimensionality of motives
– Kimura (and Jannsen); semistable vector bundles – Deninger
and Werner, and Brivio and Verra; Mordell-Weil lattices for elliptic
K3 surfaces – Shioda; and arithmetic aspects of diffraction
patterns – Stienstra. (jnek)
J. Nekovář: Selmer Complexes, Astérisque, no. 310, Société
Mathématique de France, Paris, 2006, 559 pp., EUR 86, ISBN
978-2-85629-227-3
This work gives a unified treatment of commutative Iwasawa
theory centered on a small number of sufficiently general principles.
It should be regarded as a ‘Grothendification’ of the
subject into a landscape of arithmetic geometry. The contents
of each chapter are as follows. The background material from
homological algebra is collected together in chapter 1. In chapter
2, the formalism of Grothendieck duality theory over complete
local rings is covered. Chapter 3 contains a description of
the formalism of continuous cohomology and chapter 4 deals
with finiteness results for continuous cohomology of pro-finite
groups. Results for big Galois representations from the classical
duality results for Galois cohomology of finite Galois modules
over local and global fields are deduced in chapter 5.
In chapter 6, the author introduces Selmer complexes in an
axiomatic setting and proves a duality theorem for them. A
study of generalization of unramified cohomology is contained
in chapter 7. The next two chapters contain duality results in
Iwasawa theory (deduced from those over number fields) and
their applications to classical Iwasawa theory. Various incarnations
of generalized Cassels-Tate pairings are constructed and
58 EMS Newsletter December 2007

Recent books
studied in chapter 10. The last two chapters contain a construction
of generalized p-adic height pairing and applications of ideas
developed in chapter 10 to big Galois representations arising
from Hida families of Hilbert modular forms, and to anticyclotomic
Iwasawa theory of CM points on Shimura curves. (pso)
S. P. Novikov, I. A. Taimanov: Modern Geometric Structures
and Fields, Graduate Studies in Mathematics, vol. 71, American
Mathematical Society, Providence, 2006, 633 pp., USD 79, ISBN
0-8218-3929-2
This new volume of the GSM series by the AMS has a much
wider scope than the usual textbook on differential geometry.
It covers all the basic notions (surfaces in three dimensions and
their relations to complex and conformal geometry, smooth
manifolds, groups of transformations, including crystallographic
groups, tensor algebra and tensor fields, differential forms
and their integration, the Stokes theorem and its relation to
the de Rham cohomology, connections and their curvature,
Riemannian geometry, geodesics and the Gauss-Bonnet formula).
The last third of the book treats more advanced topics
(conformal geometry, Kähler manifolds, the Morse theory,
Hamiltonian formalism, groups of symmetries and conservation
laws, Poisson and Lagrange structure on manifolds, and
multidimensional calculus of variation). The last chapter describes
basic fields of theoretical physics (Einstein’s general
relativity, Clifford algebras, spinors and the Dirac equation,
and Yang-Mills fields). The authors try to avoid unnecessary
abstraction; they present many important topics in modern geometry
and topology in a unified way. The book is designed for
students in mathematics and theoretical physics but it will be
very useful for teachers as well. (vs)
L. Nyssen, Ed.: Physics and Number Theory, IRMA Lectures in
Mathematics and Theoretical Physics, vol. 10, European Mathematical
Society, Zürich, 2006, 265 pp., EUR 39,50, ISBN 978-3-
03719-028-9
The seven articles in this stimulating collection are surveys of
a range of contemporary problems arising in physics, number
theory and the interface between the two. The topics covered
include arithmetic aspects of aperiodic crystals (articles by J.-
L. Verger-Gaugry; and J.-P. Gazeau, Z. Masáková and E. Pelantová),
phase-locking (both in classical and quantum systems) and
number theory (M. Planat), Hopf algebras in renormalization
theory (Ch. Bergbauer and D. Kreimer), L-functions and random
matrices (E. Royer), recent applications of Kloosterman
sums (P. Michel) and recent progress on the p-adic Langlands
programme (A. Mézard). The book offers interesting material
both for mathematicians and theoretical physicists. (jnek)
Séminaire Bourbaki, Volume 2004/2005, Astérisque 307, Société
Mathématique de France Paris, 2006, 520 pp., EUR 86,
ISBN 978-285629-224-2
This volume contains fourteen survey lectures: three on algebraic
geometry, two on differential geometry, one on the Poincaré
conjecture, one on dynamic systems, one on number theory,
one on the fundamentals lemma, one on the André-Oort conjecture,
one on quadratic forms, one on algebraic topology, one
on mathematical physics, and one on probabilities. As tradition
dictates, all of them contain a discussion of important achievements
(in this case in the period 2004–2005). For instance, the article
devoted to number theory reports on the Green-Tao result
on arithmetic progressions in the sequence of primes. The book
also contains contributions on the Mumford conjecture (following
the work of Madsen and Weiss), a proof of the Parisi formula
by Guerra and Talagrand, on ‘Formes quadratiques et cycles
algébriques’ (following Rost, Voevodsky, Vishik, Karpenko and
others), and so on. As one of the most important sources of reports
on major achievements in contemporary mathematics, the
volume will find a prominent place on every shelf. (spor)
J. Väänänen: Dependence Logic – A New Approach to Independence
Friendly Logic, London Mathematical Society Student
Texts 70, Cambridge University Press, Cambridge, 2007,
225 pp., GBP 23.99, ISBN 978-0-521-70015-3
This book develops and studies a conservative extension of
first order logic called dependency logic, which approaches the
study of the phenomena of dependency and independency on a
logical basis. Dependency logic introduces a new type of atomic
formula, written as = (t 1
,t 2
,…,t n
), with an intuitive meaning
that the value of the term tn depends only on the values of the
terms t 1
,t 2
,…,t n–1
. Given proper semantics, the basic language
of dependency logic is more expressive than that of first order
logic. This is demonstrated on several examples (chapter 4), e.g.
by giving a formula with no extralogical symbols whose models
are all finite sets with no infinite set. This expressiveness, of
course, cannot endure without a cost, which in the case of this
logic is the dropping of the law of the excluded middle.
The author acknowledges his inspiration as the Independence
Friendly Logic of Hintikka and Sandu, whose basic instrument
for expressing (in)dependency is instead the quantifier
“there exists xn independently of x 1
,x 2
,…,x n–1
”. While these two
logics turn out to have the same expressiveness at the level of
sentences, the author argues that dependency logic is actually
the more expressive of the two at the level of formulae (section
3.6). Model theory for dependency logic is developed by relating
the logic to existential second order logic (chapter 6). A brief yet
informative chapter 7 deals with the complexity of the decision
problem for dependency logic. The last chapter studies a further
extension of dependency logic called team logic, which besides
the non-classical negation of dependency logic also contains
Boolean negation obeying the law of the excluded middle.
The book is based on real teaching experience and it contains
many instructive exercises (for which many of the solutions are
given in an appendix by Ville Nurmi). In several places, the reader
is given the chance to acquire basic skills in a game-theoretic
approach to logic and model theory. The book is probably best
suited to advanced students of special courses but it remains accessible
to all students with a basic knowledge of logic. (ppaj)
R.E. White: Elements of Matrix Modeling and Computing
with MATLAB, Chapman & Hall/CRC, Boca Raton, 2006, 402
pp., USD 79,95, ISBN 1-58488-627-7
An important objective of this textbook is to provide “mathon-line”
for undergraduate students of science and engineering.
It is expected that students have had at least one semester of
calculus. Chapters one and two contain introductory material
on complex numbers, 2D and 3D vectors and their products. A
connection is established between geometric and algebraic approaches
to these topics. This is continued into chapters three,
four and five, where higher order algebraic systems are solved
EMS Newsletter December 2007 59

Recent books
via row operations, inverse matrices and LU decomposition.
Linearly independent vectors and subspaces are used to solve
underdetermined and overdetermined systems. Chapters six
and seven describe first and second order linear equations and
introduce eigenvalues and eigenvectors for solutions of linear
systems of initial value problems. The last two chapters use transform
methods to filter distorted images and signals. The discrete
Fourier transform is introduced via continuous versions of the
Laplace and Fourier transforms. The discrete Fourier transform
properties are derived from the Fourier matrix representation
and are used to do image filtering in the frequency domain.
Most sections have some applications, indicating that these
topics are really useful. Seven basic applications are developed
in various sections of the text, including circuits, trusses, mixing
tanks, heat conduction, data modeling and the motion of a mass
and image filters. The applications are developed from very
simple models to more complex ones. Matlab is used here to
do more complicated computations. The strategy of how to use
computing tools is given as: firstly learn the math and by-hand
calculations; secondly use computing tools to confirm by-hand
calculations; thirdly use computing tools to do more complicated
calculations and applications. (jant)
O. Zariski: The Moduli Problem for Plane Branches, University
Lecture Series, vol. 39, American Mathematical Society,
Providence, 2006, 151 pp., USD 35, ISBN 978-0-8218-2983-7
This book is the English translation of a French version published
previously by HERMANN (Éditeurs des Sciences et
des Arts, Paris). The classical problem of moduli spaces of
curves is generalized here to the problem of a description of
the moduli space of curves of the same equisingularity class.
The first four chapters are devoted to equisingularity invariants,
parametrization of the moduli space and its detailed description
(in specific cases). After a review of specific examples,
the author studies the generic component of moduli space with
a particular characteristic. The last third of the book contains
an appendix written by B. Tessier. The reader will find a version
of the ‘product decomposition’ theorem and a natural compactification
of the moduli space of a plane branch with a given
characteristic. (vs)
List of reviewers for 2007.
The Editor would like to thank the following for their reviews
this year:
J. Anděl, J. Antoch, T. Bárta, L. Barto, M. Bečvářová, L. Boček,
E. Calda, A. Drápal, M. Feistauer, S. Hencl, D. Hlubinka, Š. Holub,
J. Hurt, M. Hušek, M. Hušková, M. Hykšová, P. Kaplický, T. Kepka,
M. Klazar, P. Knobloch, J. Kofroň, J. Král, J. Lukeš, J. Málek,
J. Malý, M. Mareš, J. Milota, J. Mlček, M. Loebl, K. Najzar,
J. Nekovář, I. Netuka, J. Olejníková, P. Pajás, O. Pangrác, L. Pick,
Š. Porubský, Z. Prášková, D. Pražák, A. Procházka, P. Pyrih,
J. Rataj, M. Rokyta, T. Roubíček, P. Růžička, I. Saxl, A. Slavík,
P. Somberg, V. Souček, J. Spurný, J. Stará, J. Štěpán, J. Trlifaj,
J. Tůma, J. Vančura, J. Veselý, M. Zahradník, J. Zítko.
All of the above are on the staff of the Charles University, Faculty
of Mathematics and Physics, Prague, except: J. Vanžura (Mathematical
Institute, Czech Academy of Sciences), M. Bečvářová,
M. Hykšová (Technical University, Prague), Š. Porubský (Institute
of Computer Science, Czech Academy of Sciences) and
J. Nekovář (University Paris VI, France).
Joint EWM/EMS Worskhop
Amsterdam, July 13 th , 2008
http://womenandmath.wordpress.com/
This one day workshop, organized under
the auspices of the EMS and EWM,
aims at introducing the audience to the
topics of the two main women speakers
at the European Congress of Mathematics,
Christine Bernardi and Matilde
Marcolli. Its program will be organized around three to
four introductory talks on their research areas, and will
end with a social gathering and informal discussions. The
speakers at the workshop will include Christine Bernardi
and Alina Vdovina.
We encourage young women mathematicians to attend
this one day meeting before the Congress itself. It
will provide an opportunity to get acquainted with two
areas of research represented by two of the main speakers
at the ECM, namely applied mathematics (spectral
and variational problems) on the one hand and applications
of noncommutative geometry (e.g. to quantum field
theory and number theory) on
the other hand.
We draw your attention to
the fact that early registration at the ECM brings down
the costs (”earlybird” fee) and that you can ask for help
from the ECM to find accommodation via the university
if you apply before January 1. We encourage those of you
who can benefit from a grant delivered by the EMC, to
apply for a “one day” extension to be able to attend this
meeting.
Hoping to see you in Amsterdam,
Best wishes,
The organizers,
Colette Guillopé (Femmes et Mathématiques),
Frances Kirwan (convenor of EWM),
Sylvie Paycha (Coordinator of the EMS committee for
women in mathematics)
60 EMS Newsletter December 2007

Holomorphic Functions
in the Plane and n-
dimensional Space
Gürlebeck, K., Bauhaus University
Weimar, Germany / Habetha, K., RWTH
Aachen, Germany / Sprößig, W., TU
Freiberg, Germany
Complex analysis nowadays has higher-dimensional
analoga: the algebra of complex numbers is
replaced then by the non-commutative algebra of
real quaternions or by Clifford algebras. During the
last 30 years the so-called quaternionic and Clifford
or hypercomplex analysis successfully developed
to a powerful theory with many applications in
analysis, engineering and mathematical physics.
This textbook introduces both to classical and
higher-dimensional results based on a uniform
notion of holomorphy. Historical remarks, lots
of examples, gures and exercises accompany
each chapter. The enclosed CD-ROM contains an
extensive literature database and a Maple package
with comments and procedures of tools and
methods explained in the book.
2008. Approx. 410 pp. With CD-ROM. Softcover
EUR 34.90 / CHF 59.90
ISBN 978-3-7643-8271-1
Determinantal Ideals
Miró-Roig, R.M., Universitat de
Barcelona, Spain
Winner of the Ferran Sunyer i
Balaguer Prize 2007.
Determinantal ideals are ideals generated by minors
of a homogeneous polynomial matrix. Some classical
ideals that can be generated in this way are the ideal
of the Veronese varieties, of the Segre varieties, and
of the rational normal scrolls. Determinantal ideals
are a central topic in both commutative algebra and
algebraic geometry, and they also have numerous
connections with invariant theory, representation
theory, and combinatorics. Due to their important
role, their study has attracted many researchers and
has received considerable attention in the literature.
In this book three crucial problems are addressed:
CI-liaison class and G-liaison class of standard
determinantal ideals; the multiplicity conjecture for
standard determinantal ideals; and unobstructedness
and dimension of families of standard determinantal
ideals.
2008. Approx. 155 pp. Hardcover
EUR 39.90 / CHF 69.90 / ISBN 978-3-7643-8534-7
PM — Progress in Mathematics, Vol. 264
A History of Abstract
Algebra
Kleiner, I., York University, Toronto, ON,
Canada
Prior to the nineteenth century, algebra meant the
study of the solution of polynomial equations. By the
twentieth century it came to encompass the study of
abstract, axiomatic systems such as groups, rings, and
elds. This presentation provides an account of the
history of the basic concepts, results, and theories of
abstract algebra. The development of abstract algebra
was propelled by the need for new tools to address
certain classical problems that appeared unsolvable
by classical means. A major theme of the approach
in this book is to show how abstract algebra has
arisen in attempts to solve some of these classical
problems, providing a context from which the reader
may gain a deeper appreciation of the mathematics
involved. Mathematics instructors, algebraists, and
historians of science will nd the work a valuable
reference. The book may also serve as a supplemental
text for courses in abstract algebra or the history of
mathematics.
2007. XVI, 168 pp. 24 illus. Softcover
EUR 39.90 / CHF 65.00 / ISBN 978-0-8176-4684-4
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New Textbooks from Springer
Abstract Algebra
P. A. Grillet, Tulane University,
New Orleans, LA, USA
About the first edition
The text is geared to the
needs of the beginning graduate student, covering with
complete, well-written proofs the usual major branches
of groups, rings, fields, and modules...[n]one of the
material one expects in a book like this is missing, and
the level of detail is appropriate for its intended audience
Alberto Delgado, MathSciNet
2nd ed. 2007. XII, 676 p. (Graduate Texts in
Mathematics, Volume 242) Hardcover
ISBN 978-0-387-71567-4 € 54,95 | £42.50
The 1-2-3 of
Modular Forms
Lectures at a Summer
School in Nordfjordeid,
Norway
J. H. Bruinier, University
of Cologne, Germany;
G. van der Geer, University of
Amsterdam, The Netherlands;
G. Harder, University of Bonn, Germany; D. Zagier,
Max-Planck-Institut für Mathematik, Bonn, Germany
K. Ranestad, University of Oslo, Norway (Ed.)
This book grew out of three series of lectures given at
the summer school on “Modular Forms and their
Applications” at the Sophus Lie Conference Center in
Nordfjordeid in June 2004. Each part of the book treats
a number of beautiful applications, and together they
form a comprehensive survey for the novice and a
useful reference for a broad group of mathematicians.
2008. Approx. 230 p. (Universitext) Softcover
ISBN 978-3-540-74117-6 € 49,95 | £38.50
Probability Theory
A Comprehensive Course
2ND
EDITION
A. Klenke, Johannes Gutenberg-Universität Mainz,
Germany
Aimed primarily at graduate students and researchers,
this text is a comprehensive course in modern
probability theory and its measure-theoretical
foundations. It covers a wide variety of topics, many of
which are not usually found in introductory textbooks.
Plenty of figures, computer simulations, biographic
details of key mathematicians, and a wealth of
examples support and enliven the presentation.
2008. Approx. 630 p. 23 illus. (Universitext) Softcover
ISBN 978-1-84800-047-6 € 54,95 | £34.50
2ND
EDITION
Measure,
Topology, and
Fractal Geometry
G. Edgar, The Ohio State University,
Columbus, OH, USA
Based on a course given to talented high-school
students at Ohio University in 1988, this book is
essentially an advanced undergraduate textbook
about the mathematics of fractal geometry. It nicely
bridges the gap between traditional books on
topology/analysis and more specialized treatises on
fractal geometry. It contains plenty of examples,
exercises, and good illustrations of fractals, including
16 color plates.
From reviews of the first edition The book can be
recommended to students who seriously want to know
about the mathematical foundation of fractals, and to
lecturers who want to illustrate a standard course in
metric topology by interesting examples Christoph
Bandt, Mathematical Reviews
2nd ed. 2008. Approx. 290 p. 169 illus., 20 in color.
(Undergraduate Texts in Mathematics) Hardcover
ISBN 978-0-387-74748-4 € 39,95 | £30.50
Nodal Discontinuous
Galerkin Methods
Algorithms, Analysis,
and Applications
J. S. Hesthaven, Brown
University, Providence, RI,
USA; T. Warburton, Rice
University, Houston, TX, USA
This book introduces the key
ideas, basic analysis, and efficient implementation of
discontinuous Galerkin finite element methods (DG-
FEM) for the solution of partial differential equations.
All key theoretical results are either derived or
discussed, including an overview of relevant results
from approximation theory, convergence theory for
numerical PDE’s, orthogonal polynomials, and more.
Through embedded Matlab codes, the algorithms are
discussed and implemented for a number of classic
systems of PDEs.
2008. Approx. 290 p. (Texts in Applied Mathematics,
Volume 54) Hardcover
ISBN 978-0-387-72065-4 € 46,95 | £36.00
Putnam and
Beyond
R. Gelca, Texas Tech University,
Lubbock, TX, USA;
T. Andreescu, University of
Texas at Dallas, Richardson,
TX, USA
This book serves as a study
guide for the Putnam
Mathematics Competition and as a transition from
high school problem-solving to university level
mathematical research. Putnam and Beyond fills the
need for problem-based texts that specifically target
the Putnam exams.
2007. XVI, 798 p. 50 illus. Softcover
ISBN 978-0-387-25765-5 € 54,95 | £42.50
3RD
EDITION
Advanced Linear
Algebra
S. Roman, Irvine, CA, USA
From the reviews of the second edition This is a
formidable volume, a compendium of linear algebra
theory, classical and modern … . The development
of the subject is elegant … . The proofs are neat … .
The exercise sets are good, with occasional hints given
for the solution of trickier problems. … It represents
linear algebra and does so comprehensively Henry
Ricardo, MathDL, May, 2005)
3rd ed. 2008. XVIII, 526 p. 23 illus. (Graduate Texts in
Mathematics, Volume 135) Hardcover
ISBN 978-0-387-72828-5 € 54,95 | £42.50
Applied Linear
Algebra and
Matrix Analysis
T. S. Shores, University of
Nebraska, Lincoln, NE, USA
This new book offers a fresh
approach to matrix and
linear algebra by providing a
balanced blend of applications, theory, and computation,
while highlighting their interdependence.
Intended for a one-semester course, Applied Linear
Algebra and Matrix Analysis places special emphasis
on linear algebra as an experimental science, with
numerous examples, computer exercises, and projects.
2007. XII, 388 p. 27 illus. (Undergraduate Texts in
Mathematics) Softcover
ISBN 978-0-387-33195-9 € 32,95 | £25.50
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